---

Supervised Learning : Identifying Contributing Factors for Countries With High Cancer Rates Using Classification Algorithms With Class Imbalance Treatment

---

John Pauline Pineda

December 1, 2023

---

• 1. Table of Contents
  • 1.1 Introduction
    • 1.1.1 Study Objectives
    • 1.1.2 Outcome
    • 1.1.3 Predictors
  • 1.2 Methodology
    • 1.2.1 Data Assessment
    • 1.2.2 Feature Selection
    • 1.2.3 Model Formulation
    • 1.2.4 Model Hyperparameter Tuning
    • 1.2.5 Model Performance Evaluation
    • 1.2.6 Model Presentation
  • 1.3 Results
    • 1.3.1 Data Preparation
    • 1.3.2 Data Quality Assessment
    • 1.3.3 Data Preprocessing
      • 1.3.3.1 Data Cleaning
      • 1.3.3.2 Missing Data Imputation
      • 1.3.3.3 Outlier Treatment
      • 1.3.3.4 Collinearity
      • 1.3.3.5 Shape Transformation
      • 1.3.3.6 Centering and Scaling
      • 1.3.3.7 Data Encoding
      • 1.3.3.8 Preprocessed Data Description
    • 1.3.4 Data Exploration
      • 1.3.4.1 Exploratory Data Analysis
      • 1.3.4.2 Hypothesis Testing
    • 1.3.5 Model Development With Hyperparameter Tuning
      • 1.3.5.1 Premodelling Data Description
      • 1.3.5.2 Logistic Regression
      • 1.3.5.3 Decision Trees
      • 1.3.5.4 Random Forest
      • 1.3.5.5 Support Vector Machine
    • 1.3.6 Model Development With Class Weights
      • 1.3.6.1 Premodelling Data Description
      • 1.3.6.2 Logistic Regression
      • 1.3.6.3 Decision Trees
      • 1.3.6.4 Random Forest
      • 1.3.6.5 Support Vector Machine
    • 1.3.7 Model Development With SMOTE Upsampling
      • 1.3.7.1 Premodelling Data Description
      • 1.3.7.2 Logistic Regression
      • 1.3.7.3 Decision Trees
      • 1.3.7.4 Random Forest
      • 1.3.7.5 Support Vector Machine
    • 1.3.8 Model Development With CNN Downsampling
      • 1.3.8.1 Premodelling Data Description
      • 1.3.8.2 Logistic Regression
      • 1.3.8.3 Decision Trees
      • 1.3.8.4 Random Forest
      • 1.3.8.5 Support Vector Machine
    • 1.3.9 Model Development With Stacking Ensemble Learning
      • 1.3.9.1 Premodelling Data Description
      • 1.3.9.2 Logistic Regression
    • 1.3.10 Model Selection
    • 1.3.11 Model Presentation
      • 1.3.11.1 Odds Ratio
      • 1.3.11.2 Shapley Additive Explanations
• 2. Summary
• 3. References

---

1. Table of Contents

1.1 Introduction

Age-standardized cancer rates are measures used to compare cancer incidence between countries while accounting for differences in age distribution. They allow for a more accurate assessment of the relative risk of cancer across populations with diverse demographic and socio-economic characteristics - enabling a more nuanced understanding of the global burden of cancer and facilitating evidence-based public health interventions.

Datasets used for the analysis were separately gathered and consolidated from various sources including:

1. Cancer Rates from World Population Review
2. Social Protection and Labor Indicator from World Bank
3. Education Indicator from World Bank
4. Economy and Growth Indicator from World Bank
5. Environment Indicator from World Bank
6. Climate Change Indicator from World Bank
7. Agricultural and Rural Development Indicator from World Bank
8. Social Development Indicator from World Bank
9. Health Indicator from World Bank
10. Science and Technology Indicator from World Bank
11. Urban Development Indicator from World Bank
12. Human Development Indices from Human Development Reports
13. Environmental Performance Indices from Yale Center for Environmental Law and Policy

This study hypothesized that various global development indicators and indices influence cancer rates across countries.

Subsequent analysis and modelling steps involving data understanding, data preparation, data exploration, model development, model validation and model presentation were individually detailed below, with all the results consolidated in a Summary provided at the end of the document.

1.1.1 Study Objectives

The main objective of the study is to develop an interpretable classification model which could provide robust and reliable predictions of belonging to a group of countries with high cancer rates from an optimal set of observations and predictors, while addressing class imbalance and delivering accurate predictions when applied to new unseen data.

Specific objectives are given as follows:

• Obtain an optimal subset of observations and predictors by conducting data quality assessment and feature selection, excluding cases or variables noted with irregularities and applying preprocessing operations most suitable for the downstream analysis

• Develop multiple classification models with remedial measures applied to address class imbalance and with optimized hyperparameters through internal resampling validation

• Select the final classification model among candidates based on robust performance estimates

• Evaluate the final model performance and generalization ability through external validation in an independent set

• Conduct a post-hoc exploration of the model results to provide general insights on the importance, contribution and effect of the various predictors to model prediction

1.1.2 Outcome

The analysis endpoint for the study is described below:

• CANRAT (categorical): Age-standardized cancer rates of countries dichotomized to two categories pertaining to those classified in the upper 25th percentile and lower 75th percentile, per 100K population (2022)

1.1.3 Predictors

Detailed descriptions for each individual predictor used in the study are provided as follows:

• GDPPER (numeric): GDP per person employed, current US Dollars (2020)
• URBPOP (numeric): Urban population, % of total population (2020)
• PATRES (numeric): Patent applications by residents, total count (2020)
• RNDGDP (numeric): Research and development expenditure, % of GDP (2020)
• POPGRO (numeric): Population growth, annual % (2020)
• LIFEXP (numeric): Life expectancy at birth, total in years (2020)
• TUBINC (numeric): Incidence of tuberculosis, per 100K population (2020)
• DTHCMD (numeric): Cause of death by communicable diseases and maternal, prenatal and nutrition conditions, % of total (2019)
• AGRLND (numeric): Agricultural land, % of land area (2020)
• GHGEMI (numeric): Total greenhouse gas emissions, kt of CO2 equivalent (2020)
• RELOUT (numeric): Renewable electricity output, % of total electricity output (2015)
• METEMI (numeric): Methane emissions, kt of CO2 equivalent (2020)
• FORARE (numeric): Forest area, % of land area (2020)
• CO2EMI (numeric): CO2 emissions, metric tons per capita (2020)
• PM2EXP (numeric): PM2.5 air pollution, population exposed to levels exceeding WHO guideline value, % of total (2017)
• POPDEN (numeric): Population density, people per sq. km of land area (2020)
• GDPCAP (numeric): GDP per capita, current US Dollars (2020)
• ENRTER (numeric): Tertiary school enrollment, % gross (2020)
• HDICAT (categorical): Human development index, ordered category (2020)
• EPISCO (numeric): Environment performance index , score (2022)

1.2 Methodology

1.2.1 Data Assessment

Preliminary data used in the study was evaluated and prepared for analysis and modelling using the following methods:

Data Quality Assessment involves profiling and assessing the data to understand its suitability for machine learning tasks. The quality of training data has a huge impact on the efficiency, accuracy and complexity of machine learning tasks. Data remains susceptible to errors or irregularities that may be introduced during collection, aggregation or annotation stage. Issues such as incorrect labels, synonymous categories in a categorical variable or heterogeneity in columns, among others, which might go undetected by standard pre-processing modules in these frameworks can lead to sub-optimal model performance, inaccurate analysis and unreliable decisions.

Data Preprocessing involves changing the raw feature vectors into a representation that is more suitable for the downstream modelling and estimation processes, including data cleaning, integration, reduction and transformation. Data cleaning aims to identify and correct errors in the dataset that may negatively impact a predictive model such as removing outliers, replacing missing values, smoothing noisy data, and correcting inconsistent data. Data integration addresses potential issues with redundant and inconsistent data obtained from multiple sources through approaches such as detection of tuple duplication and data conflict. The purpose of data reduction is to have a condensed representation of the data set that is smaller in volume, while maintaining the integrity of the original data set. Data transformation converts the data into the most appropriate form for data modeling.

Data Exploration involves analyzing and investigating data sets to summarize their main characteristics, often employing data visualization methods. It helps determine how best to manipulate data sources to discover patterns, spot anomalies, test a hypothesis, or check assumptions. This process is primarily used to see what data can reveal beyond the formal modeling or hypothesis testing task and provides a better understanding of data set variables and the relationships between them.

Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to imputation of the target columns. The Linear Regression model was formulated for imputation - which explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.

1.2.2 Feature Selection

Statistical test measures were assessed for the numeric and categorical predictors in the study to determine the most optimal subset of variables for the subsequent modelling process which included the following:

Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.

Two-Sample T-Test Statistic is used to determine whether there is a significant difference between the means of two independent groups. It is calculated as the difference between the means of the two groups divided by the standard error of the difference. The test statistic follows a t-distribution with degrees of freedom calculated based on the sample sizes and assumptions about the variances.

Chi-square Test Statistic is used to assess whether there is a significant association between two categorical variables. It is calculated by comparing the observed frequencies of the contingency table with the frequencies that would be expected if the variables were independent. The test statistic follows a chi-square distribution, and the degrees of freedom are determined by the number of categories in the variables being analyzed.

1.2.3 Model Formulation

This study implemented both glass-box and black-box classification modelling procedures with simple to complex structures involving moderate to large numbers of model coefficients or mathematical transformations which lacked transparency in terms of the internal processes and weighted factors used in reaching a decision. Models applied in the analysis for predicting the categorical target were the following:

Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.

Decision Trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)

Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.

Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.

Different versions of the individual models were formulated following remedial measures to address class imbalance described as follows:

Hyperparameter Tuning is an iterative process that involves experimenting with different hyperparameter combinations, evaluating the model's performance, and refining the hyperparameter values to achieve the best possible performance on new, unseen data - aimed at building effective and well-generalizing machine learning models. A model's performance depends not only on the learned parameters (weights) during training but also on hyperparameters, which are external configuration settings that cannot be learned from the data.

Class Weights are used to assign different levels of importance to different classes when the distribution of instances across different classes in a classification problem is not equal. By assigning higher weights to the minority class, the model is encouraged to give more attention to correctly predicting instances from the minority class. Class weights are incorporated into the loss function during training. The loss for each instance is multiplied by its corresponding class weight. This means that misclassifying an instance from the minority class will have a greater impact on the overall loss than misclassifying an instance from the majority class. The use of class weights helps balance the influence of each class during training, mitigating the impact of class imbalance. It provides a way to focus the learning process on the classes that are underrepresented in the training data.

Synthetic Minority Oversampling Technique is specifically designed to increase the representation of the minority class by generating new minority instances between existing instances. The new instances created are not just the copy of existing minority cases, instead for each minority class instance, the algorithm generates synthetic examples by creating linear combinations of the feature vectors between that instance and its k nearest neighbors. The synthetic samples are placed along the line segments connecting the original instance to its neighbors.

Condensed Nearest Neighbors is a prototype selection algorithm that aims to select a subset of instances from the original dataset, discarding redundant and less informative instances. The algorithm works by iteratively adding instances to the subset, starting with an empty set. At each iteration, an instance is added if it is not correctly classified by the current subset. The decision to add or discard an instance is based on its performance on a k-nearest neighbors classifier. If an instance is misclassified by the current subset's k-nearest neighbors, it is added to the subset. The process is repeated until no new instances are added to the subset. The resulting subset is a condensed representation of the dataset that retains the essential information needed for classification.

An additional iteration of the modelling process applying an ensemble structure was carried out for comparison:

Model Stacking - also known as stacked generalization, is an ensemble approach which involves creating a variety of base learners and using them to create intermediate predictions, one for each learned model. A meta-model is incorporated that gains knowledge of the same target from intermediate predictions. Unlike bagging, in stacking, the models are typically different (e.g. not all decision trees) and fit on the same dataset (e.g. instead of samples of the training dataset). Unlike boosting, in stacking, a single model is used to learn how to best combine the predictions from the contributing models (e.g. instead of a sequence of models that correct the predictions of prior models). Stacking is appropriate when the predictions made by the base learners or the errors in predictions made by the models have minimal correlation. Achieving an improvement in performance is dependent upon the choice of base learners and whether they are sufficiently skillful in their predictions.

1.2.4 Model Hyperparameter Tuning

The optimal combination of hyperparameter values which maximized the performance of the various classification models in the study used the following hyperparameter tuning strategy:

K-Fold Cross-Validation involves dividing the training set after a random shuffle into a user-defined K number of smaller non-overlapping sets called folds. Each unique fold is assigned as the hold-out test data to assess the model trained from the data set collected from all the remaining K-1 folds. The evaluation score is retained but the model is discarded. The process is recursively performed resulting to a total of K fitted models and evaluated on the K hold-out test sets. All K-computed performance measures reported from the process are then averaged to represent the estimated performance of the model. This approach can be computationally expensive and may be highly dependent on how the data was randomly assigned to their respective folds, but does not waste too much data which is a major advantage in problems where the number of samples is very small.

1.2.5 Model Performance Evaluation

The predictive performance of the formulated classification models in the study were compared and evaluated using the following metrics:

Accuracy is the ratio of correctly predicted instances to the total instances. It provides an overall measure of model performance which is easy to understand and interpret, but can be misleading in imbalanced datasets when one class dominates.

Precision is the ratio of correctly predicted positive observations to the total predicted positives. It is useful when the cost of false positives is high but does not consider false negatives, so might not be suitable for imbalanced datasets.

Recall is the ratio of correctly predicted positive observations to all the actual positives. It is useful when the cost of false negatives is high but does not consider false positives, so might not be suitable for imbalanced datasets.

F1 Score is the harmonic mean of precision and recall. It balances precision and recall, providing a single metric for performance evaluation which is suitable for imbalanced datasets.Although, it might not be the best metric in situations where precision or recall is more critical.

AUROC measures the area under the receiver operating characteristic curve, which illustrates the trade-off between true positive rate (sensitivity) and false positive rate at various classification thresholds. It provides a comprehensive evaluation of the model's ability to discriminate between classes and is robust to imbalanced datasets. Compared to other metrics, it may not be as directly interpretable as well as not being sensitive to class distribution changes.

1.2.6 Model Presentation

Model presentation was conducted post-hoc and focused on both model-specific and model-agnostic techniques which did not consider any assumptions about the model structures. These methods were described as follows:

Odds Ratios aid in interpreting the relationship between the independent variables and the probability of an event occurring in a logistic regression model by quantifying the change in odds associated with a one-unit change in the independent variable. An estimated value greater than one indicates that the odds of the event are expected to increase by a factor equal to the odds ratio for a one-unit increase in the independent variable. While a an estimated value less than one indicates that The odds of the event are expected to decrease by the reciprocal of the odds ratio for a one-unit increase in the independent variable.

Shapley Additive Explanations are based on Shapley values developed in the cooperative game theory. The process involves explaining a prediction by assuming that each explanatory variable for an instance is a player in a game where the prediction is the payout. The game is the prediction task for a single instance of the data set. The gain is the actual prediction for this instance minus the average prediction for all instances. The players are the explanatory variable values of the instance that collaborate to receive the gain (predict a certain value). The determined value is the average marginal contribution of an explanatory variable across all possible coalitions.

1.3. Results

1.3.1. Data Preparation

1. The initial tabular dataset was comprised of 177 observations and 22 variables (including 1 metadata, 1 target and 20 predictors).
   • 177 rows (observations)
   • 22 columns (variables)
     • 1/22 metadata (object)
       • COUNTRY
     • 1/22 target (categorical)
       • CANRAT
     • 19/22 predictor (numeric)
       • GDPPER
       • URBPOP
       • PATRES
       • RNDGDP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • RELOUT
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • ENRTER
       • EPISCO
     • 1/22 predictor (categorical)
       • HDICAT

##################################
# Setting up compatibility issues
# between the scikit-learn and imblearn packages
##################################
# !pip uninstall scikit-learn --yes
# !pip uninstall imblearn --yes
# !pip install scikit-learn==1.2.2
# !pip install imblearn
##################################
# Installing shap package
##################################
# !pip install shap

##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
%matplotlib inline
import shap

from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import StandardScaler
from scipy import stats

from sklearn.linear_model import LogisticRegression
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import RandomForestClassifier
from sklearn.svm import SVC
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score, roc_auc_score, confusion_matrix, ConfusionMatrixDisplay
from sklearn.model_selection import train_test_split, GridSearchCV
from imblearn.over_sampling import SMOTE
from imblearn.under_sampling import CondensedNearestNeighbour
from sklearn.ensemble import StackingClassifier

##################################
# Loading the dataset
##################################
cancer_rate = pd.read_csv('CategoricalCancerRates.csv')

##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)

Column Names and Data Types:

COUNTRY     object
CANRAT      object
GDPPER     float64
URBPOP     float64
PATRES     float64
RNDGDP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
RELOUT     float64
METEMI     float64
FORARE     float64
CO2EMI     float64
PM2EXP     float64
POPDEN     float64
ENRTER     float64
GDPCAP     float64
HDICAT      object
EPISCO     float64
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()

	COUNTRY	CANRAT	GDPPER	URBPOP	PATRES	RNDGDP	POPGRO	LIFEXP	TUBINC	DTHCMD	...	RELOUT	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	ENRTER	GDPCAP	HDICAT	EPISCO
0	Australia	High	98380.63601	86.241	2368.0	NaN	1.235701	83.200000	7.2	4.941054	...	13.637841	131484.763200	17.421315	14.772658	24.893584	3.335312	110.139221	51722.06900	VH	60.1
1	New Zealand	High	77541.76438	86.699	348.0	NaN	2.204789	82.256098	7.2	4.354730	...	80.081439	32241.937000	37.570126	6.160799	NaN	19.331586	75.734833	41760.59478	VH	56.7
2	Ireland	High	198405.87500	63.653	75.0	1.23244	1.029111	82.556098	5.3	5.684596	...	27.965408	15252.824630	11.351720	6.768228	0.274092	72.367281	74.680313	85420.19086	VH	57.4
3	United States	High	130941.63690	82.664	269586.0	3.42287	0.964348	76.980488	2.3	5.302060	...	13.228593	748241.402900	33.866926	13.032828	3.343170	36.240985	87.567657	63528.63430	VH	51.1
4	Denmark	High	113300.60110	88.116	1261.0	2.96873	0.291641	81.602439	4.1	6.826140	...	65.505925	7778.773921	15.711000	4.691237	56.914456	145.785100	82.664330	60915.42440	VH	77.9

5 rows × 22 columns

##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].astype('category')
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].cat.set_categories(['Low', 'High'], ordered=True)
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)

##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())

Numeric Variable Summary:

	count	mean	std	min	25%	50%	75%	max
GDPPER	165.0	45284.424283	3.941794e+04	1718.804896	13545.254510	34024.900890	66778.416050	2.346469e+05
URBPOP	174.0	59.788121	2.280640e+01	13.345000	42.432750	61.701500	79.186500	1.000000e+02
PATRES	108.0	20607.388889	1.340683e+05	1.000000	35.250000	244.500000	1297.750000	1.344817e+06
RNDGDP	74.0	1.197474	1.189956e+00	0.039770	0.256372	0.873660	1.608842	5.354510e+00
POPGRO	174.0	1.127028	1.197718e+00	-2.079337	0.236900	1.179959	2.031154	3.727101e+00
LIFEXP	174.0	71.746113	7.606209e+00	52.777000	65.907500	72.464610	77.523500	8.456000e+01
TUBINC	174.0	105.005862	1.367229e+02	0.770000	12.000000	44.500000	147.750000	5.920000e+02
DTHCMD	170.0	21.260521	1.927333e+01	1.283611	6.078009	12.456279	36.980457	6.520789e+01
AGRLND	174.0	38.793456	2.171551e+01	0.512821	20.130276	40.386649	54.013754	8.084112e+01
GHGEMI	170.0	259582.709895	1.118550e+06	179.725150	12527.487367	41009.275980	116482.578575	1.294287e+07
RELOUT	153.0	39.760036	3.191492e+01	0.000296	10.582691	32.381668	63.011450	1.000000e+02
METEMI	170.0	47876.133575	1.346611e+05	11.596147	3662.884908	11118.976025	32368.909040	1.186285e+06
FORARE	173.0	32.218177	2.312001e+01	0.008078	11.604388	31.509048	49.071780	9.741212e+01
CO2EMI	170.0	3.751097	4.606479e+00	0.032585	0.631924	2.298368	4.823496	3.172684e+01
PM2EXP	167.0	91.940595	2.206003e+01	0.274092	99.627134	100.000000	100.000000	1.000000e+02
POPDEN	174.0	200.886765	6.453834e+02	2.115134	27.454539	77.983133	153.993650	7.918951e+03
ENRTER	116.0	49.994997	2.970619e+01	2.432581	22.107195	53.392460	71.057467	1.433107e+02
GDPCAP	170.0	13992.095610	1.957954e+04	216.827417	1870.503029	5348.192875	17421.116227	1.173705e+05
EPISCO	165.0	42.946667	1.249086e+01	18.900000	33.000000	40.900000	50.500000	7.790000e+01

##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())

Object Variable Summary:

	count	unique	top	freq
COUNTRY	177	177	Australia	1

##################################
# Performing a general exploration of the categorical variables
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())

Categorical Variable Summary:

	count	unique	top	freq
CANRAT	177	2	Low	132
HDICAT	167	4	VH	59

##################################
# Performing a general exploration of the categorical variable
##################################
cancer_rate.HDICAT.value_counts(normalize = True)

VH    0.353293
H     0.233533
M     0.221557
L     0.191617
Name: HDICAT, dtype: float64

##################################
# Performing a general exploration of the response variable
##################################
cancer_rate.CANRAT.value_counts(normalize = True)

Low     0.745763
High    0.254237
Name: CANRAT, dtype: float64

1.3.2 Data Quality Assessment

Data quality findings based on assessment are as follows:

1. No duplicated rows observed.
2. Missing data noted for 20 variables with Null.Count>0 and Fill.Rate<1.0.
   • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
   • PATRES: Null.Count = 69, Fill.Rate = 0.610
   • ENRTER: Null.Count = 61, Fill.Rate = 0.655
   • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • GDPPER: Null.Count = 12, Fill.Rate = 0.932
   • EPISCO: Null.Count = 12, Fill.Rate = 0.932
   • HDICAT: Null.Count = 10, Fill.Rate = 0.943
   • PM2EXP: Null.Count = 10, Fill.Rate = 0.943
   • DTHCMD: Null.Count = 7, Fill.Rate = 0.960
   • METEMI: Null.Count = 7, Fill.Rate = 0.960
   • CO2EMI: Null.Count = 7, Fill.Rate = 0.960
   • GDPCAP: Null.Count = 7, Fill.Rate = 0.960
   • GHGEMI: Null.Count = 7, Fill.Rate = 0.960
   • FORARE: Null.Count = 4, Fill.Rate = 0.977
   • TUBINC: Null.Count = 3, Fill.Rate = 0.983
   • AGRLND: Null.Count = 3, Fill.Rate = 0.983
   • POPGRO: Null.Count = 3, Fill.Rate = 0.983
   • POPDEN: Null.Count = 3, Fill.Rate = 0.983
   • URBPOP: Null.Count = 3, Fill.Rate = 0.983
   • LIFEXP: Null.Count = 3, Fill.Rate = 0.983
3. 120 observations noted with at least 1 missing data. From this number, 14 observations reported high Missing.Rate>0.2.
   • COUNTRY=Guadeloupe: Missing.Rate= 0.909
   • COUNTRY=Martinique: Missing.Rate= 0.909
   • COUNTRY=French Guiana: Missing.Rate= 0.909
   • COUNTRY=New Caledonia: Missing.Rate= 0.500
   • COUNTRY=French Polynesia: Missing.Rate= 0.500
   • COUNTRY=Guam: Missing.Rate= 0.500
   • COUNTRY=Puerto Rico: Missing.Rate= 0.409
   • COUNTRY=North Korea: Missing.Rate= 0.227
   • COUNTRY=Somalia: Missing.Rate= 0.227
   • COUNTRY=South Sudan: Missing.Rate= 0.227
   • COUNTRY=Venezuela: Missing.Rate= 0.227
   • COUNTRY=Libya: Missing.Rate= 0.227
   • COUNTRY=Eritrea: Missing.Rate= 0.227
   • COUNTRY=Yemen: Missing.Rate= 0.227
4. Low variance observed for 1 variable with First.Second.Mode.Ratio>5.
   • PM2EXP: First.Second.Mode.Ratio = 53.000
5. No low variance observed for any variable with Unique.Count.Ratio>10.
6. High skewness observed for 5 variables with Skewness>3 or Skewness<(-3).
   • POPDEN: Skewness = +10.267
   • GHGEMI: Skewness = +9.496
   • PATRES: Skewness = +9.284
   • METEMI: Skewness = +5.801
   • PM2EXP: Skewness = -3.141

##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()

0

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                              data_type_list,
                                              row_count_list,
                                              non_null_count_list,
                                              null_count_list,
                                              fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(all_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	COUNTRY	object	177	177	0	1.000000
1	CANRAT	category	177	177	0	1.000000
2	GDPPER	float64	177	165	12	0.932203
3	URBPOP	float64	177	174	3	0.983051
4	PATRES	float64	177	108	69	0.610169
5	RNDGDP	float64	177	74	103	0.418079
6	POPGRO	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051
9	DTHCMD	float64	177	170	7	0.960452
10	AGRLND	float64	177	174	3	0.983051
11	GHGEMI	float64	177	170	7	0.960452
12	RELOUT	float64	177	153	24	0.864407
13	METEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
15	CO2EMI	float64	177	170	7	0.960452
16	PM2EXP	float64	177	167	10	0.943503
17	POPDEN	float64	177	174	3	0.983051
18	ENRTER	float64	177	116	61	0.655367
19	GDPCAP	float64	177	170	7	0.960452
20	HDICAT	category	177	167	10	0.943503
21	EPISCO	float64	177	165	12	0.932203

##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])

20

##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
5	RNDGDP	float64	177	74	103	0.418079
4	PATRES	float64	177	108	69	0.610169
18	ENRTER	float64	177	116	61	0.655367
12	RELOUT	float64	177	153	24	0.864407
2	GDPPER	float64	177	165	12	0.932203
21	EPISCO	float64	177	165	12	0.932203
20	HDICAT	category	177	167	10	0.943503
16	PM2EXP	float64	177	167	10	0.943503
9	DTHCMD	float64	177	170	7	0.960452
13	METEMI	float64	177	170	7	0.960452
15	CO2EMI	float64	177	170	7	0.960452
19	GDPCAP	float64	177	170	7	0.960452
11	GHGEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
8	TUBINC	float64	177	174	3	0.983051
10	AGRLND	float64	177	174	3	0.983051
6	POPGRO	float64	177	174	3	0.983051
17	POPDEN	float64	177	174	3	0.983051
3	URBPOP	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051

##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]

##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()

##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))

##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))

##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)

##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
                                           column_count_list,
                                           null_row_list,
                                           missing_rate_list), 
                                        columns=['Row.Name',
                                                 'Column.Count',
                                                 'Null.Count',                                                 
                                                 'Missing.Rate'])
display(all_row_quality_summary)

	Row.Name	Column.Count	Null.Count	Missing.Rate
0	Australia	22	1	0.045455
1	New Zealand	22	2	0.090909
2	Ireland	22	0	0.000000
3	United States	22	0	0.000000
4	Denmark	22	0	0.000000
...	...	...	...	...
172	Congo Republic	22	3	0.136364
173	Bhutan	22	2	0.090909
174	Nepal	22	2	0.090909
175	Gambia	22	4	0.181818
176	Niger	22	2	0.090909

177 rows × 4 columns

##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])

120

##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])

14

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))

	Row.Name	Column.Count	Null.Count	Missing.Rate
35	Guadeloupe	22	20	0.909091
39	Martinique	22	20	0.909091
56	French Guiana	22	20	0.909091
13	New Caledonia	22	11	0.500000
44	French Polynesia	22	11	0.500000
75	Guam	22	11	0.500000
53	Puerto Rico	22	9	0.409091
85	North Korea	22	6	0.272727
132	Somalia	22	6	0.272727
168	South Sudan	22	6	0.272727
73	Venezuela	22	5	0.227273
117	Libya	22	5	0.227273
161	Eritrea	22	5	0.227273
164	Yemen	22	5	0.227273

##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns

##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()

##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()

##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()

##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()

##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]

##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]

##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)

##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)

##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))

##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()

##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()

numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                numeric_minimum_list,
                                                numeric_mean_list,
                                                numeric_median_list,
                                                numeric_maximum_list,
                                                numeric_first_mode_list,
                                                numeric_second_mode_list,
                                                numeric_first_mode_count_list,
                                                numeric_second_mode_count_list,
                                                numeric_first_second_mode_ratio_list,
                                                numeric_unique_count_list,
                                                numeric_row_count_list,
                                                numeric_unique_count_ratio_list,
                                                numeric_skewness_list,
                                                numeric_kurtosis_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Minimum',
                                                 'Mean',
                                                 'Median',
                                                 'Maximum',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio',
                                                 'Skewness',
                                                 'Kurtosis'])
display(numeric_column_quality_summary)

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
0	GDPPER	1718.804896	45284.424283	34024.900890	2.346469e+05	98380.636010	42154.178100	1	1	1.000000	165	177	0.932203	1.517574	3.471992
1	URBPOP	13.345000	59.788121	61.701500	1.000000e+02	100.000000	52.516000	2	1	2.000000	173	177	0.977401	-0.210702	-0.962847
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
3	RNDGDP	0.039770	1.197474	0.873660	5.354510e+00	1.232440	0.962180	1	1	1.000000	74	177	0.418079	1.396742	1.695957
4	POPGRO	-2.079337	1.127028	1.179959	3.727101e+00	1.235701	1.483129	1	1	1.000000	174	177	0.983051	-0.195161	-0.423580
5	LIFEXP	52.777000	71.746113	72.464610	8.456000e+01	83.200000	68.687000	1	1	1.000000	174	177	0.983051	-0.357965	-0.649601
6	TUBINC	0.770000	105.005862	44.500000	5.920000e+02	12.000000	7.200000	4	3	1.333333	131	177	0.740113	1.746333	2.429368
7	DTHCMD	1.283611	21.260521	12.456279	6.520789e+01	4.941054	42.079403	1	1	1.000000	170	177	0.960452	0.900509	-0.691541
8	AGRLND	0.512821	38.793456	40.386649	8.084112e+01	46.252480	72.006469	1	1	1.000000	174	177	0.983051	0.074000	-0.926249
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	3000.932259	1	1	1.000000	170	177	0.960452	9.496120	101.637308
10	RELOUT	0.000296	39.760036	32.381668	1.000000e+02	100.000000	13.637841	3	1	3.000000	151	177	0.853107	0.501088	-0.981774
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	1326.034028	1	1	1.000000	170	177	0.960452	5.801014	38.661386
12	FORARE	0.008078	32.218177	31.509048	9.741212e+01	17.421315	8.782159	1	1	1.000000	173	177	0.977401	0.519277	-0.322589
13	CO2EMI	0.032585	3.751097	2.298368	3.172684e+01	14.772658	0.972088	1	1	1.000000	170	177	0.960452	2.721552	10.311574
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	13.300785	1	1	1.000000	174	177	0.983051	10.267750	119.995256
16	ENRTER	2.432581	49.994997	53.392460	1.433107e+02	110.139221	45.220661	1	1	1.000000	116	177	0.655367	0.275863	-0.392895
17	GDPCAP	216.827417	13992.095610	5348.192875	1.173705e+05	51722.069000	3961.726633	1	1	1.000000	170	177	0.960452	2.258568	5.938690
18	EPISCO	18.900000	42.946667	40.900000	7.790000e+01	29.600000	43.600000	3	3	1.000000	137	177	0.774011	0.641799	0.035208

##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])

1

##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
14	PM2EXP	0.274092	91.940595	100.0	100.0	100.0	100.0	106	2	53.0	61	177	0.344633	-3.141557	9.032386

##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])

5

##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	13.300785	1	1	1.000000	174	177	0.983051	10.267750	119.995256
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	3000.932259	1	1	1.000000	170	177	0.960452	9.496120	101.637308
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	1326.034028	1	1	1.000000	170	177	0.960452	5.801014	38.661386
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386

##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')

##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns

##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]

##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]

##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]

##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]

##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)

##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)

##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))

##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)

object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
                                                 object_first_mode_list,
                                                 object_second_mode_list,
                                                 object_first_mode_count_list,
                                                 object_second_mode_count_list,
                                                 object_first_second_mode_ratio_list,
                                                 object_unique_count_list,
                                                 object_row_count_list,
                                                 object_unique_count_ratio_list), 
                                        columns=['Object.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(object_column_quality_summary)

	Object.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	COUNTRY	Australia	Mauritius	1	1	1.0	177	177	1.0

##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')

##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns

##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]

##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]

##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)

##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)

##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)

categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
                                                    categorical_first_mode_list,
                                                    categorical_second_mode_list,
                                                    categorical_first_mode_count_list,
                                                    categorical_second_mode_count_list,
                                                    categorical_first_second_mode_ratio_list,
                                                    categorical_unique_count_list,
                                                    categorical_row_count_list,
                                                    categorical_unique_count_ratio_list), 
                                        columns=['Categorical.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(categorical_column_quality_summary)

	Categorical.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	CANRAT	Low	High	132	45	2.933333	2	177	0.011299
1	HDICAT	VH	H	59	39	1.512821	4	177	0.022599

##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])

0

1.3.3. Data Preprocessing

1.3.3.1 Data Cleaning

1. Subsets of rows and columns with high rates of missing data were removed from the dataset:
   • 4 variables with Fill.Rate<0.9 were excluded for subsequent analysis.
     • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
     • PATRES: Null.Count = 69, Fill.Rate = 0.610
     • ENRTER: Null.Count = 61, Fill.Rate = 0.655
     • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • 14 rows with Missing.Rate>0.2 were exluded for subsequent analysis.
     • COUNTRY=Guadeloupe: Missing.Rate= 0.909
     • COUNTRY=Martinique: Missing.Rate= 0.909
     • COUNTRY=French Guiana: Missing.Rate= 0.909
     • COUNTRY=New Caledonia: Missing.Rate= 0.500
     • COUNTRY=French Polynesia: Missing.Rate= 0.500
     • COUNTRY=Guam: Missing.Rate= 0.500
     • COUNTRY=Puerto Rico: Missing.Rate= 0.409
     • COUNTRY=North Korea: Missing.Rate= 0.227
     • COUNTRY=Somalia: Missing.Rate= 0.227
     • COUNTRY=South Sudan: Missing.Rate= 0.227
     • COUNTRY=Venezuela: Missing.Rate= 0.227
     • COUNTRY=Libya: Missing.Rate= 0.227
     • COUNTRY=Eritrea: Missing.Rate= 0.227
     • COUNTRY=Yemen: Missing.Rate= 0.227
2. No variables were removed due to zero or near-zero variance.
3. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 15/18 predictor (numeric)
       • GDPPER
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/18 predictor (categorical)
       • HDICAT

##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)

Dataset Dimensions: 

(163, 22)

##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)

##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)

Dataset Dimensions: 

(163, 18)

1.3.3.2 Missing Data Imputation

1. Missing data for numeric variables were imputed using the iterative imputer algorithm with a linear regression estimator.
   • GDPPER: Null.Count = 1
   • FORARE: Null.Count = 1
   • PM2EXP: Null.Count = 5
2. Missing data for categorical variables were imputed using the most frequent value.
   • HDICAP: Null.Count = 1

##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
                                                  list(cancer_rate_cleaned.dtypes),
                                                  list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
                                                  list(cancer_rate_cleaned.count()),
                                                  list(cancer_rate_cleaned.isna().sum(axis=0))), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count'])
display(cleaned_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count
0	COUNTRY	object	163	163	0
1	CANRAT	category	163	163	0
2	GDPPER	float64	163	162	1
3	URBPOP	float64	163	163	0
4	POPGRO	float64	163	163	0
5	LIFEXP	float64	163	163	0
6	TUBINC	float64	163	163	0
7	DTHCMD	float64	163	163	0
8	AGRLND	float64	163	163	0
9	GHGEMI	float64	163	163	0
10	METEMI	float64	163	163	0
11	FORARE	float64	163	162	1
12	CO2EMI	float64	163	163	0
13	PM2EXP	float64	163	158	5
14	POPDEN	float64	163	163	0
15	GDPCAP	float64	163	163	0
16	HDICAT	category	163	162	1
17	EPISCO	float64	163	163	0

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')

##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')

##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	NaN	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()

##################################
# Defining the parameter of the
# iterative imputer which will estimate 
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
    estimator = lr,
    max_iter = 10,
    tol = 1e-10,
    imputation_order = 'ascending',
    random_state=88888888
)

##################################
# Implementing the iterative imputer 
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)

##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array, 
                                           columns = cancer_rate_cleaned_numeric.columns)

##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	65.867296	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')

##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0], inplace=True)
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)

##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')  

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                                  data_type_list,
                                                  row_count_list,
                                                  non_null_count_list,
                                                  null_count_list,
                                                  fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(imputed_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	GDPPER	float64	163	163	0	1.0
1	URBPOP	float64	163	163	0	1.0
2	POPGRO	float64	163	163	0	1.0
3	LIFEXP	float64	163	163	0	1.0
4	TUBINC	float64	163	163	0	1.0
5	DTHCMD	float64	163	163	0	1.0
6	AGRLND	float64	163	163	0	1.0
7	GHGEMI	float64	163	163	0	1.0
8	METEMI	float64	163	163	0	1.0
9	FORARE	float64	163	163	0	1.0
10	CO2EMI	float64	163	163	0	1.0
11	PM2EXP	float64	163	163	0	1.0
12	POPDEN	float64	163	163	0	1.0
13	GDPCAP	float64	163	163	0	1.0
14	EPISCO	float64	163	163	0	1.0
15	CANRAT	category	163	163	0	1.0
16	HDICAT	category	163	163	0	1.0

1.3.3.3 Outlier Detection

1. High number of outliers observed for 5 numeric variables with Outlier.Ratio>0.10 and marginal to high Skewness.
   • PM2EXP: Outlier.Count = 37, Outlier.Ratio = 0.226, Skewness=-3.061
   • GHGEMI: Outlier.Count = 27, Outlier.Ratio = 0.165, Skewness=+9.299
   • GDPCAP: Outlier.Count = 22, Outlier.Ratio = 0.134, Skewness=+2.311
   • POPDEN: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+9.972
   • METEMI: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+5.688
2. Minimal number of outliers observed for 5 numeric variables with Outlier.Ratio<0.10 and normal Skewness.
   • TUBINC: Outlier.Count = 12, Outlier.Ratio = 0.073, Skewness=+1.747
   • CO2EMI: Outlier.Count = 11, Outlier.Ratio = 0.067, Skewness=+2.693
   • GDPPER: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+1.554
   • EPISCO: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+0.635
   • CANRAT: Outlier.Count = 2, Outlier.Ratio = 0.012, Skewness=+0.910

##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()

##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1

##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()

##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)

##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                  numeric_skewness_list,
                                                  numeric_outlier_count_list,
                                                  numeric_row_count_list,
                                                  numeric_outlier_ratio_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Skewness',
                                                 'Outlier.Count',
                                                 'Row.Count',
                                                 'Outlier.Ratio'])
display(numeric_column_outlier_summary)

	Numeric.Column.Name	Skewness	Outlier.Count	Row.Count	Outlier.Ratio
0	GDPPER	1.554457	3	163	0.018405
1	URBPOP	-0.212327	0	163	0.000000
2	POPGRO	-0.181666	0	163	0.000000
3	LIFEXP	-0.329704	0	163	0.000000
4	TUBINC	1.747962	12	163	0.073620
5	DTHCMD	0.930709	0	163	0.000000
6	AGRLND	0.035315	0	163	0.000000
7	GHGEMI	9.299960	27	163	0.165644
8	METEMI	5.688689	20	163	0.122699
9	FORARE	0.563015	0	163	0.000000
10	CO2EMI	2.693585	11	163	0.067485
11	PM2EXP	-3.088403	37	163	0.226994
12	POPDEN	9.972806	20	163	0.122699
13	GDPCAP	2.311079	22	163	0.134969
14	EPISCO	0.635994	3	163	0.018405

##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_imputed_numeric, x=column)

1.3.3.4 Collinearity

1. Majority of the numeric variables reported moderate to high correlation which were statistically significant.
2. Among pairwise combinations of numeric variables, high Pearson.Correlation.Coefficient values were noted for:
   • GDPPER and GDPCAP: Pearson.Correlation.Coefficient = +0.921
   • GHGEMI and METEMI: Pearson.Correlation.Coefficient = +0.905
3. Among the highly correlated pairs, variables with the lowest correlation against the target variable were removed.
   • GDPPER: Pearson.Correlation.Coefficient = +0.690
   • METEMI: Pearson.Correlation.Coefficient = +0.062
4. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 16 columns (variables)
     • 1/16 metadata (object)
       • COUNTRY
     • 1/16 target (categorical)
       • CANRAT
     • 13/16 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/16 predictor (categorical)
       • HDICAT

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
    f, ax = plt.subplots(figsize=(11, 9))
    sns.heatmap(corr, 
                ax=ax,
                mask=mask,
                annot=True, 
                vmin=-1, 
                vmax=1, 
                center=0,
                cmap='coolwarm', 
                linewidths=1, 
                linecolor='gray', 
                cbar_kws={'orientation': 'horizontal'})  

##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
    cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
        cancer_rate_imputed_numeric.loc[:, numeric_column_a], 
        cancer_rate_imputed_numeric.loc[:, numeric_column_b])

##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))

	Pearson.Correlation.Coefficient	Correlation.PValue
GDPPER_GDPCAP	0.921010	8.158179e-68
GHGEMI_METEMI	0.905121	1.087643e-61
POPGRO_DTHCMD	0.759470	7.124695e-32
GDPPER_LIFEXP	0.755787	2.055178e-31
GDPCAP_EPISCO	0.696707	5.312642e-25
LIFEXP_GDPCAP	0.683834	8.321371e-24
GDPPER_EPISCO	0.680812	1.555304e-23
GDPPER_URBPOP	0.666394	2.781623e-22
GDPPER_CO2EMI	0.654958	2.450029e-21
TUBINC_DTHCMD	0.643615	1.936081e-20
URBPOP_LIFEXP	0.623997	5.669778e-19
LIFEXP_EPISCO	0.620271	1.048393e-18
URBPOP_GDPCAP	0.559181	8.624533e-15
CO2EMI_GDPCAP	0.550221	2.782997e-14
URBPOP_CO2EMI	0.550046	2.846393e-14
LIFEXP_CO2EMI	0.531305	2.951829e-13
URBPOP_EPISCO	0.510131	3.507463e-12
POPGRO_TUBINC	0.442339	3.384403e-09
DTHCMD_PM2EXP	0.283199	2.491837e-04
CO2EMI_EPISCO	0.282734	2.553620e-04

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
    p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
    for col in df.columns:
        for col2 in df.drop(col,axis=1).columns:
            _ , p = stats.pearsonr(df[col],df[col2])
            p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
    return p_matrix

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)                     
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05)) 
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)  

##################################
# Filtering out one among the 
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)

Dataset Dimensions: 

(163, 13)

1.3.3.5 Shape Transformation

1. A Yeo-Johnson transformation was applied to all numeric variables to improve distributional shape.
2. Most variables achieved symmetrical distributions with minimal outliers after transformation.
3. One variable which remained skewed even after applying shape transformation was removed.
   • PM2EXP
4. The transformed dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
                                          standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)

##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
                                               columns=cancer_rate_imputed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_transformed_numeric, x=column)

##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)

Dataset Dimensions: 

(163, 12)

1.3.3.6 Centering and Scaling

1. All numeric variables were transformed using the standardization method to achieve a comparable scale between values.
2. The scaled dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting standardization
# to transform the values of the 
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)

##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
                                          columns=cancer_rate_transformed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_scaled_numeric, x=column)

1.3.3.7 Data Encoding

1. One-hot encoding was applied to the HDICAP_VH variable resulting to 4 additional columns in the dataset:
   • HDICAP_L
   • HDICAP_M
   • HDICAP_H
   • HDICAP_VH

##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
                                               columns=['HDICAT'])

##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])

1.3.3.8 Preprocessed Data Description

1. The preprocessed dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 12/18 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 4/18 predictor (categorical)
       • HDICAT_L
       • HDICAT_M
       • HDICAT_H
       • HDICAT_VH

##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')  

##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)

Dataset Dimensions: 

(163, 16)

1.3.4 Data Exploration

1.3.4.1 Exploratory Data Analysis

1. Bivariate analysis identified individual predictors with generally positive association to the target variable based on visual inspection.
2. Higher values or higher proportions for the following predictors are associated with the CANRAT HIGH category:
   • URBPOP
   • LIFEXP
   • CO2EMI
   • GDPCAP
   • EPISCO
   • HDICAP_VH=1
3. Decreasing values or smaller proportions for the following predictors are associated with the CANRAT LOW category:
   • POPGRO
   • TUBINC
   • DTHCMD
   • HDICAP_L=0
   • HDICAP_M=0
   • HDICAP_H=0
4. Values for the following predictors are not associated with the CANRAT HIGH or LOW categories:
   • AGRLND
   • GHGEMI
   • FORARE
   • POPDEN

##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = cancer_rate_filtered_row['CANRAT'].to_frame()
cancer_rate_preprocessed_target.reset_index(inplace=True, drop=True)
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[cancer_rate_categorical_encoded.columns]
cancer_rate_preprocessed_categorical_combined = cancer_rate_preprocessed_categorical.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed = cancer_rate_preprocessed.drop(cancer_rate_categorical_encoded.columns, axis=1) 
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.columns
cancer_rate_preprocessed_combined = cancer_rate_preprocessed.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed_all = cancer_rate_preprocessed_combined.join(cancer_rate_categorical_encoded)

##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 6
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 30))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual boxplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
    ax = axes[i]
    ax.boxplot([group[x_variable] for name, group in cancer_rate_preprocessed_combined.groupby(y_variable)])
    ax.set_title(f'{y_variable} Versus {x_variable}')
    ax.set_xlabel(y_variable)
    ax.set_ylabel(x_variable)
    ax.set_xticks(range(1, len(cancer_rate_preprocessed_combined[y_variable].unique()) + 1), ['Low', 'High'])

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

##################################
# Segregating the target
# and predictor variable names
##################################
y_variables = cancer_rate_preprocessed_categorical.columns
x_variable = 'CANRAT'

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 2
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 10))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual stacked column plots
# for all categorical columns
##################################
for i, y_variable in enumerate(y_variables):
    ax = axes[i]
    category_counts = cancer_rate_preprocessed_categorical_combined.groupby([x_variable, y_variable]).size().unstack(fill_value=0)
    category_proportions = category_counts.div(category_counts.sum(axis=1), axis=0)
    category_proportions.plot(kind='bar', stacked=True, ax=ax)
    ax.set_title(f'{x_variable} Versus {y_variable}')
    ax.set_xlabel(x_variable)
    ax.set_ylabel('Proportions')

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

1.3.4.2 Hypothesis Testing

1. The relationship between the numeric predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: Difference in the means between groups LOW and HIGH is equal to zero
   • Alternative: Difference in the means between groups LOW and HIGH is not equal to zero
2. There is sufficient evidence to conclude of a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable in 9 of the 12 numeric predictors given their high t-test statistic values with reported low p-values less than the significance level of 0.05.
   • GDPCAP: T.Test.Statistic=-11.937, T.Test.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, T.Test.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, T.Test.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, T.Test.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, T.Test.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, T.Test.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, T.Test.PValue=0.000
   • POPGRO: T.Test.Statistic=+4.905, T.Test.PValue=0.000
   • GHGEMI: T.Test.Statistic=-2.243, T.Test.PValue=0.026
3. The relationship between the categorical predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: The categorical predictor is independent of the categorical target variable
   • Alternative: The categorical predictor is dependent of the categorical target variable
4. There is sufficient evidence to conclude of a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable in all 4 categorical predictors given their high chisquare statistic values with reported low p-values less than the significance level of 0.05.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
   • HDICAT_H: ChiSquare.Test.Statistic=13.860, ChiSquare.Test.PValue=0.000
   • HDICAT_M: ChiSquare.Test.Statistic=10.286, ChiSquare.Test.PValue=0.001
   • HDICAT_L: ChiSquare.Test.Statistic=9.081, ChiSquare.Test.PValue=0.002

##################################
# Computing the t-test 
# statistic and p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_ttest_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed_combined
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_predictors
for numeric_column in cancer_rate_preprocessed_numeric_columns:
    group_0 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='Low']
    group_1 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='High']
    cancer_rate_preprocessed_numeric_ttest_target['CANRAT_' + numeric_column] = stats.ttest_ind(
        group_0[numeric_column], 
        group_1[numeric_column], 
        equal_var=True)

##################################
# Formulating the pairwise ttest summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_ttest_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(12))

	T.Test.Statistic	T.Test.PValue
CANRAT_GDPCAP	-11.936988	6.247937e-24
CANRAT_EPISCO	-11.788870	1.605980e-23
CANRAT_LIFEXP	-10.979098	2.754214e-21
CANRAT_TUBINC	9.608760	1.463678e-17
CANRAT_DTHCMD	8.375558	2.552108e-14
CANRAT_CO2EMI	-7.030702	5.537463e-11
CANRAT_URBPOP	-6.541001	7.734940e-10
CANRAT_POPGRO	4.904817	2.269446e-06
CANRAT_GHGEMI	-2.243089	2.625563e-02
CANRAT_FORARE	-1.174143	2.420717e-01
CANRAT_POPDEN	-0.495221	6.211191e-01
CANRAT_AGRLND	-0.047628	9.620720e-01

##################################
# Computing the chisquare
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_chisquare_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed_categorical_combined
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
    contingency_table = pd.crosstab(cancer_rate_preprocessed_categorical[categorical_column], 
                                    cancer_rate_preprocessed_categorical['CANRAT'])
    cancer_rate_preprocessed_categorical_chisquare_target['CANRAT_' + categorical_column] = stats.chi2_contingency(
        contingency_table)[0:2]

##################################
# Formulating the pairwise chisquare summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_chisquare_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['ChiSquare.Test.Statistic', 'ChiSquare.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['ChiSquare.Test.PValue'], ascending=True).head(4))

	ChiSquare.Test.Statistic	ChiSquare.Test.PValue
CANRAT_HDICAT_VH	76.764134	1.926446e-18
CANRAT_HDICAT_M	13.860367	1.969074e-04
CANRAT_HDICAT_L	10.285575	1.340742e-03
CANRAT_HDICAT_H	9.080788	2.583087e-03

1.3.5. Model Development With Hyperparameter Tuning

1.3.5.1 Premodelling Data Description

1. Among the 9 numeric variables determined to have a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable, only 7 were retained with absolute T-Test statistics greater than 5.
   • GDPCAP: T.Test.Statistic=-11.937, T.Test.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, T.Test.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, T.Test.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, T.Test.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, T.Test.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, T.Test.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, T.Test.PValue=0.000
2. Among the 4 categorical predictors determined to have a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable, only 1 was retained with absolute Chi-Square statistics greater than 15.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 9)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP        float64
LIFEXP        float64
TUBINC        float64
DTHCMD        float64
CO2EMI        float64
GDPCAP        float64
EPISCO        float64
CANRAT       category
HDICAT_VH       uint8
dtype: object

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling,
             kind='reg',
             hue='CANRAT');
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 8)

##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)

0    0.745614
1    0.254386
dtype: float64

##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)

Dataset Dimensions: 

(49, 8)

##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)

0    0.755102
1    0.244898
dtype: float64

##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
    metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
    metric_value = [accuracy_score(y_true, y_pred),
                   precision_score(y_true, y_pred),
                   recall_score(y_true, y_pred),
                   f1_score(y_true, y_pred),
                   roc_auc_score(y_true, y_pred)]    
    metric_summary = pd.DataFrame(zip(metric_name, metric_value),
                                  columns=['metric_name','metric_value']) 
    return(metric_summary)

1.3.5.2 Logistic Regression

1. The logistic regression model from the sklearn.linear_model Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • penalty = penalty norm made to vary between L1 and L2
   • solver = algorithm used in the optimization problem made to vary between Saga and Liblinear
   • class_weight = weights associated with classes held constant at a value of None
   • max_iter = maximum number of iterations taken for the solvers to converge held constant at a value of 500
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • penalty = L1 norm
   • solver = Liblinear
   • class_weight = None
   • max_iter = 500
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9473
   • Precision = 0.8709
   • Recall = 0.9310
   • F1 Score = 0.9000
   • AUROC = 0.9419
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.8889
   • Recall = 0.6667
   • F1 Score = 0.7619
   • AUROC = 0.8198
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'penalty': ['l1', 'l2'],
    'solver': ['liblinear','saga'],
    'class_weight': [None],
    'max_iter': [500],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
optimal_logistic_regression = GridSearchCV(estimator = logistic_regression, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Logistic Regression model
##################################
optimal_logistic_regression.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Logistic Regression model
##################################
optimal_logistic_regression.best_score_ 
optimal_logistic_regression.best_params_

{'C': 1.0,
 'class_weight': None,
 'max_iter': 500,
 'penalty': 'l1',
 'random_state': 88888888,
 'solver': 'liblinear'}

##################################
# Evaluating the optimal Logistic Regression model
# on the train set
##################################
optimal_logistic_regression_y_hat_train = optimal_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_logistic_regression_performance_train = model_performance_evaluation(y_train, optimal_logistic_regression_y_hat_train)
optimal_logistic_regression_performance_train['model'] = ['optimal_logistic_regression'] * 5
optimal_logistic_regression_performance_train['set'] = ['train'] * 5
print('Optimal Logistic Regression Model Performance on Train Data: ')
display(optimal_logistic_regression_performance_train)

Optimal Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	optimal_logistic_regression	train
1	Precision	0.870968	optimal_logistic_regression	train
2	Recall	0.931034	optimal_logistic_regression	train
3	F1	0.900000	optimal_logistic_regression	train
4	AUROC	0.941988	optimal_logistic_regression	train

##################################
# Evaluating the optimal Logistic Regression model
# on the test set
##################################
optimal_logistic_regression_y_hat_test = optimal_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_logistic_regression_performance_test = model_performance_evaluation(y_test, optimal_logistic_regression_y_hat_test)
optimal_logistic_regression_performance_test['model'] = ['optimal_logistic_regression'] * 5
optimal_logistic_regression_performance_test['set'] = ['test'] * 5
print('Optimal Logistic Regression Model Performance on Test Data: ')
display(optimal_logistic_regression_performance_test)

Optimal Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	optimal_logistic_regression	test
1	Precision	0.888889	optimal_logistic_regression	test
2	Recall	0.666667	optimal_logistic_regression	test
3	F1	0.761905	optimal_logistic_regression	test
4	AUROC	0.819820	optimal_logistic_regression	test

1.3.5.3 Decision Trees

1. The decision tree model from the sklearn.tree Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • class_weight = weights associated with classes held constant at a value of None
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Entropy
   • max_depth = 5
   • min_samples_leaf = 3
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9736
   • Precision = 1.0000
   • Recall = 0.8965
   • F1 Score = 0.9454
   • AUROC = 0.9482
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8571
   • Precision = 0.8571
   • Recall = 0.5000
   • F1 Score = 0.6315
   • AUROC = 0.7364
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
optimal_decision_tree = GridSearchCV(estimator = decision_tree, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Decision Tree model
##################################
optimal_decision_tree.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Decision Tree model
##################################
optimal_decision_tree.best_score_ 
optimal_decision_tree.best_params_

{'class_weight': None,
 'criterion': 'entropy',
 'max_depth': 5,
 'min_samples_leaf': 3,
 'random_state': 88888888}

##################################
# Evaluating the optimal decision tree model
# on the train set
##################################
optimal_decision_tree_y_hat_train = optimal_decision_tree.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_decision_tree_performance_train = model_performance_evaluation(y_train, optimal_decision_tree_y_hat_train)
optimal_decision_tree_performance_train['model'] = ['optimal_decision_tree'] * 5
optimal_decision_tree_performance_train['set'] = ['train'] * 5
print('Optimal Decision Tree Model Performance on Train Data: ')
display(optimal_decision_tree_performance_train)

Optimal Decision Tree Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.973684	optimal_decision_tree	train
1	Precision	1.000000	optimal_decision_tree	train
2	Recall	0.896552	optimal_decision_tree	train
3	F1	0.945455	optimal_decision_tree	train
4	AUROC	0.948276	optimal_decision_tree	train

##################################
# Evaluating the optimal decision tree model
# on the test set
##################################
optimal_decision_tree_y_hat_test = optimal_decision_tree.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_decision_tree_performance_test = model_performance_evaluation(y_test, optimal_decision_tree_y_hat_test)
optimal_decision_tree_performance_test['model'] = ['optimal_decision_tree'] * 5
optimal_decision_tree_performance_test['set'] = ['test'] * 5
print('Optimal Decision Tree Model Performance on Test Data: ')
display(optimal_decision_tree_performance_test)

Optimal Decision Tree Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.857143	optimal_decision_tree	test
1	Precision	0.857143	optimal_decision_tree	test
2	Recall	0.500000	optimal_decision_tree	test
3	F1	0.631579	optimal_decision_tree	test
4	AUROC	0.736486	optimal_decision_tree	test

1.3.5.4 Random Forest

1. The random forest model from the sklearn.ensemble Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • n_estimators = number of trees in the forest made to vary between 100, 150 and 200
   • max_features = number of features to consider when looking for the best split made to vary between Sqrt and Log2 of n_estimators
   • class_weight = weights associated with classes held constant at a value of None
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Gini
   • max_depth = 3
   • min_samples_leaf = 3
   • n_estimators = 100
   • max_features = Sqrt n_estimators
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9561
   • Precision = 0.9285
   • Recall = 0.8965
   • F1 Score = 0.9122
   • AUROC = 0.9365
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8775
   • Precision = 0.8750
   • Recall = 0.5833
   • F1 Score = 0.7000
   • AUROC = 0.7781
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Random Forest model
##################################
random_forest = RandomForestClassifier()

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'n_estimators': [100,150,200],
    'max_features':['sqrt', 'log2'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
optimal_random_forest = GridSearchCV(estimator = random_forest, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Random Forest model
##################################
optimal_random_forest.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Random Forest model
##################################
optimal_random_forest.best_score_ 
optimal_random_forest.best_params_

{'class_weight': None,
 'criterion': 'gini',
 'max_depth': 3,
 'max_features': 'sqrt',
 'min_samples_leaf': 3,
 'n_estimators': 100,
 'random_state': 88888888}

##################################
# Evaluating the optimal Random Forest model
# on the train set
##################################
optimal_random_forest_y_hat_train = optimal_random_forest.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_random_forest_performance_train = model_performance_evaluation(y_train, optimal_random_forest_y_hat_train)
optimal_random_forest_performance_train['model'] = ['optimal_random_forest'] * 5
optimal_random_forest_performance_train['set'] = ['train'] * 5
print('Optimal Random Forest Model Performance on Train Data: ')
display(optimal_random_forest_performance_train)

Optimal Random Forest Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.956140	optimal_random_forest	train
1	Precision	0.928571	optimal_random_forest	train
2	Recall	0.896552	optimal_random_forest	train
3	F1	0.912281	optimal_random_forest	train
4	AUROC	0.936511	optimal_random_forest	train

##################################
# Evaluating the optimal Random Forest model
# on the test set
##################################
optimal_random_forest_y_hat_test = optimal_random_forest.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_random_forest_performance_test = model_performance_evaluation(y_test, optimal_random_forest_y_hat_test)
optimal_random_forest_performance_test['model'] = ['optimal_random_forest'] * 5
optimal_random_forest_performance_test['set'] = ['test'] * 5
print('Optimal Random Forest Model Performance on Test Data: ')
display(optimal_random_forest_performance_test)

Optimal Random Forest Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.877551	optimal_random_forest	test
1	Precision	0.875000	optimal_random_forest	test
2	Recall	0.583333	optimal_random_forest	test
3	F1	0.700000	optimal_random_forest	test
4	AUROC	0.778153	optimal_random_forest	test

1.3.5.5 Support Vector Machine

1. The support vector machine model from the sklearn.svm Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • kernel = kernel type to be used in the algorithm made to vary between Linear, Poly, RBF and Sigmoid
   • class_weight = weights associated with classes held constant at a value of None
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • kernel = Poly
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9473
   • Precision = 0.9600
   • Recall = 0.8275
   • F1 Score = 0.8888
   • AUROC = 0.9079
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8571
   • Precision = 0.8571
   • Recall = 0.5000
   • F1 Score = 0.6315
   • AUROC = 0.7364
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Support Vector Machine model
##################################
support_vector_machine = SVC()

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
optimal_support_vector_machine = GridSearchCV(estimator = support_vector_machine, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Support Vector Machine model
##################################
optimal_support_vector_machine.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Support Vector Machine model
##################################
optimal_support_vector_machine.best_score_ 
optimal_support_vector_machine.best_params_

{'C': 1.0, 'class_weight': None, 'kernel': 'poly', 'random_state': 88888888}

##################################
# Evaluating the optimal Support Vector Machine model
# on the train set
##################################
optimal_support_vector_machine_y_hat_train = optimal_support_vector_machine.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_support_vector_machine_performance_train = model_performance_evaluation(y_train, optimal_support_vector_machine_y_hat_train)
optimal_support_vector_machine_performance_train['model'] = ['optimal_support_vector_machine'] * 5
optimal_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Optimal Support Vector Machine Model Performance on Train Data: ')
display(optimal_support_vector_machine_performance_train)

Optimal Support Vector Machine Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	optimal_support_vector_machine	train
1	Precision	0.960000	optimal_support_vector_machine	train
2	Recall	0.827586	optimal_support_vector_machine	train
3	F1	0.888889	optimal_support_vector_machine	train
4	AUROC	0.907911	optimal_support_vector_machine	train

##################################
# Evaluating the optimal Support Vector Machine model
# on the test set
##################################
optimal_support_vector_machine_y_hat_test = optimal_support_vector_machine.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_support_vector_machine_performance_test = model_performance_evaluation(y_test, optimal_support_vector_machine_y_hat_test)
optimal_support_vector_machine_performance_test['model'] = ['optimal_support_vector_machine'] * 5
optimal_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Optimal Support Vector Machine Model Performance on Test Data: ')
display(optimal_support_vector_machine_performance_test)

Optimal Support Vector Machine Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.857143	optimal_support_vector_machine	test
1	Precision	0.857143	optimal_support_vector_machine	test
2	Recall	0.500000	optimal_support_vector_machine	test
3	F1	0.631579	optimal_support_vector_machine	test
4	AUROC	0.736486	optimal_support_vector_machine	test

1.3.6 Model Development With Class Weights

1.3.6.1 Premodelling Data Description

1. Among the 9 numeric variables determined to have a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable, only 7 were retained with absolute T-Test statistics greater than 5.
   • GDPCAP: T.Test.Statistic=-11.937, T.Test.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, T.Test.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, T.Test.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, T.Test.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, T.Test.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, T.Test.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, T.Test.PValue=0.000
2. Among the 4 categorical predictors determined to have a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable, only 1 was retained with absolute Chi-Square statistics greater than 15.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 9)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP        float64
LIFEXP        float64
TUBINC        float64
DTHCMD        float64
CO2EMI        float64
GDPCAP        float64
EPISCO        float64
CANRAT       category
HDICAT_VH       uint8
dtype: object

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling,
             kind='reg',
             hue='CANRAT');
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 8)

##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)

0    0.745614
1    0.254386
dtype: float64

##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)

Dataset Dimensions: 

(49, 8)

##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)

0    0.755102
1    0.244898
dtype: float64

##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
    metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
    metric_value = [accuracy_score(y_true, y_pred),
                   precision_score(y_true, y_pred),
                   recall_score(y_true, y_pred),
                   f1_score(y_true, y_pred),
                   roc_auc_score(y_true, y_pred)]    
    metric_summary = pd.DataFrame(zip(metric_name, metric_value),
                                  columns=['metric_name','metric_value']) 
    return(metric_summary)

1.3.6.2 Logistic Regression

1. The logistic regression model from the sklearn.linear_model Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • penalty = penalty norm made to vary between L1 and L2
   • solver = algorithm used in the optimization problem made to vary between Saga and Liblinear
   • class_weight = weights associated with classes held constant at a value of 25-75 between classes 0 and 1
   • max_iter = maximum number of iterations taken for the solvers to converge held constant at a value of 500
3. The original data reflecting a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • penalty = L1 norm
   • solver = Liblinear
   • class_weight = 25-75 between classes 0 and 1
   • max_iter = 500
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.8947
   • Precision = 0.7073
   • Recall = 1.0000
   • F1 Score = 0.8285
   • AUROC = 0.9294
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.9387
   • Precision = 0.8461
   • Recall = 0.9167
   • F1 Score = 0.8800
   • AUROC = 0.9313
7. Considerable difference in the apparent and independent test model performance observed, indicative of the presence of moderate model overfitting.

##################################
# Creating an instance of the 
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'penalty': ['l1', 'l2'],
    'solver': ['liblinear','saga'],
    'class_weight': [{0:0.25, 1:0.75}],
    'max_iter': [500],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
weighted_logistic_regression = GridSearchCV(estimator = logistic_regression, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Logistic Regression model
##################################
weighted_logistic_regression.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Logistic Regression model
##################################
weighted_logistic_regression.best_score_ 
weighted_logistic_regression.best_params_

{'C': 1.0,
 'class_weight': {0: 0.25, 1: 0.75},
 'max_iter': 500,
 'penalty': 'l2',
 'random_state': 88888888,
 'solver': 'liblinear'}

##################################
# Evaluating the weighted Logistic Regression model
# on the train set
##################################
weighted_logistic_regression_y_hat_train = weighted_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_logistic_regression_performance_train = model_performance_evaluation(y_train, weighted_logistic_regression_y_hat_train)
weighted_logistic_regression_performance_train['model'] = ['weighted_logistic_regression'] * 5
weighted_logistic_regression_performance_train['set'] = ['train'] * 5
print('Weighted Logistic Regression Model Performance on Train Data: ')
display(weighted_logistic_regression_performance_train)

Weighted Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.894737	weighted_logistic_regression	train
1	Precision	0.707317	weighted_logistic_regression	train
2	Recall	1.000000	weighted_logistic_regression	train
3	F1	0.828571	weighted_logistic_regression	train
4	AUROC	0.929412	weighted_logistic_regression	train

##################################
# Evaluating the weighted Logistic Regression model
# on the test set
##################################
weighted_logistic_regression_y_hat_test = weighted_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_logistic_regression_performance_test = model_performance_evaluation(y_test, weighted_logistic_regression_y_hat_test)
weighted_logistic_regression_performance_test['model'] = ['weighted_logistic_regression'] * 5
weighted_logistic_regression_performance_test['set'] = ['test'] * 5
print('Weighted Logistic Regression Model Performance on Test Data: ')
display(weighted_logistic_regression_performance_test)

Weighted Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.938776	weighted_logistic_regression	test
1	Precision	0.846154	weighted_logistic_regression	test
2	Recall	0.916667	weighted_logistic_regression	test
3	F1	0.880000	weighted_logistic_regression	test
4	AUROC	0.931306	weighted_logistic_regression	test

1.3.6.3 Decision Trees

1. The decision tree model from the sklearn.tree Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • class_weight = weights associated with classes held constant at a value of 25-75 between classes 0 and 1
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Gini
   • max_depth = 3
   • min_samples_leaf = 3
   • class_weight = 25-75 between classes 0 and 1
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9736
   • Precision = 1.0000
   • Recall = 0.8965
   • F1 Score = 0.9454
   • AUROC = 0.9482
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8571
   • Precision = 0.8571
   • Recall = 0.5000
   • F1 Score = 0.6315
   • AUROC = 0.7364
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
weighted_decision_tree = GridSearchCV(estimator = decision_tree, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Decision Tree model
##################################
weighted_decision_tree.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Decision Tree model
##################################
weighted_decision_tree.best_score_ 
weighted_decision_tree.best_params_

{'class_weight': {0: 0.25, 1: 0.75},
 'criterion': 'gini',
 'max_depth': 3,
 'min_samples_leaf': 3,
 'random_state': 88888888}

##################################
# Evaluating the weighted decision tree model
# on the train set
##################################
weighted_decision_tree_y_hat_train = weighted_decision_tree.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_decision_tree_performance_train = model_performance_evaluation(y_train, weighted_decision_tree_y_hat_train)
weighted_decision_tree_performance_train['model'] = ['weighted_decision_tree'] * 5
weighted_decision_tree_performance_train['set'] = ['train'] * 5
print('Weighted Decision Tree Model Performance on Train Data: ')
display(weighted_decision_tree_performance_train)

Weighted Decision Tree Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.956140	weighted_decision_tree	train
1	Precision	0.852941	weighted_decision_tree	train
2	Recall	1.000000	weighted_decision_tree	train
3	F1	0.920635	weighted_decision_tree	train
4	AUROC	0.970588	weighted_decision_tree	train

##################################
# Evaluating the weighted decision tree model
# on the test set
##################################
weighted_decision_tree_y_hat_test = weighted_decision_tree.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_decision_tree_performance_test = model_performance_evaluation(y_test, weighted_decision_tree_y_hat_test)
weighted_decision_tree_performance_test['model'] = ['weighted_decision_tree'] * 5
weighted_decision_tree_performance_test['set'] = ['test'] * 5
print('Weighted Decision Tree Model Performance on Test Data: ')
display(weighted_decision_tree_performance_test)

Weighted Decision Tree Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	weighted_decision_tree	test
1	Precision	0.769231	weighted_decision_tree	test
2	Recall	0.833333	weighted_decision_tree	test
3	F1	0.800000	weighted_decision_tree	test
4	AUROC	0.876126	weighted_decision_tree	test

1.3.6.4 Random Forest

1. The random forest model from the sklearn.ensemble Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • n_estimators = number of trees in the forest made to vary between 100, 150 and 200
   • max_features = number of features to consider when looking for the best split made to vary between Sqrt and Log2 of n_estimators
   • class_weight = weights associated with classes held constant at a value of 25-75 between classes 0 and 1
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Gini
   • max_depth = 5
   • min_samples_leaf = 3
   • n_estimators = 100
   • max_features = Sqrt n_estimators
   • class_weight = 25-75 between classes 0 and 1
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9736
   • Precision = 0.9062
   • Recall = 1.0000
   • F1 Score = 0.9508
   • AUROC = 0.9823
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.8888
   • Recall = 0.6666
   • F1 Score = 0.7619
   • AUROC = 0.8198
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Random Forest model
##################################
random_forest = RandomForestClassifier()

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'n_estimators': [100,150,200],
    'max_features':['sqrt', 'log2'],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
weighted_random_forest = GridSearchCV(estimator = random_forest, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Random Forest model
##################################
weighted_random_forest.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Random Forest model
##################################
weighted_random_forest.best_score_ 
weighted_random_forest.best_params_

{'class_weight': {0: 0.25, 1: 0.75},
 'criterion': 'gini',
 'max_depth': 5,
 'max_features': 'sqrt',
 'min_samples_leaf': 3,
 'n_estimators': 100,
 'random_state': 88888888}

##################################
# Evaluating the weighted Random Forest model
# on the train set
##################################
weighted_random_forest_y_hat_train = weighted_random_forest.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_random_forest_performance_train = model_performance_evaluation(y_train, weighted_random_forest_y_hat_train)
weighted_random_forest_performance_train['model'] = ['weighted_random_forest'] * 5
weighted_random_forest_performance_train['set'] = ['train'] * 5
print('Weighted Random Forest Model Performance on Train Data: ')
display(weighted_random_forest_performance_train)

Weighted Random Forest Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.973684	weighted_random_forest	train
1	Precision	0.906250	weighted_random_forest	train
2	Recall	1.000000	weighted_random_forest	train
3	F1	0.950820	weighted_random_forest	train
4	AUROC	0.982353	weighted_random_forest	train

##################################
# Evaluating the weighted Random Forest model
# on the test set
##################################
weighted_random_forest_y_hat_test = weighted_random_forest.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_random_forest_performance_test = model_performance_evaluation(y_test, weighted_random_forest_y_hat_test)
weighted_random_forest_performance_test['model'] = ['weighted_random_forest'] * 5
weighted_random_forest_performance_test['set'] = ['test'] * 5
print('Weighted Random Forest Model Performance on Test Data: ')
display(weighted_random_forest_performance_test)

Weighted Random Forest Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	weighted_random_forest	test
1	Precision	0.888889	weighted_random_forest	test
2	Recall	0.666667	weighted_random_forest	test
3	F1	0.761905	weighted_random_forest	test
4	AUROC	0.819820	weighted_random_forest	test

1.3.6.5 Support Vector Machine

1. The support vector machine model from the sklearn.svm Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • kernel = kernel type to be used in the algorithm made to vary between Linear, Poly, RBF and Sigmoid
   • class_weight = weights associated with classes held constant at a value of 25-75 between classes 0 and 1
3. The original data which reflect a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • kernel = Poly
   • class_weight = 25-75 between classes 0 and 1
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9649
   • Precision = 0.9629
   • Recall = 0.8965
   • F1 Score = 0.9285
   • AUROC = 0.9423
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8775
   • Precision = 0.8750
   • Recall = 0.5833
   • F1 Score = 0.7000
   • AUROC = 0.7781
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Support Vector Machine model
##################################
support_vector_machine = SVC()

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
weighted_support_vector_machine = GridSearchCV(estimator = support_vector_machine, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Support Vector Machine model
##################################
weighted_support_vector_machine.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Support Vector Machine model
##################################
weighted_support_vector_machine.best_score_ 
weighted_support_vector_machine.best_params_

{'C': 1.0,
 'class_weight': {0: 0.25, 1: 0.75},
 'kernel': 'poly',
 'random_state': 88888888}

##################################
# Evaluating the weighted Support Vector Machine model
# on the train set
##################################
weighted_support_vector_machine_y_hat_train = weighted_support_vector_machine.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_support_vector_machine_performance_train = model_performance_evaluation(y_train, weighted_support_vector_machine_y_hat_train)
weighted_support_vector_machine_performance_train['model'] = ['weighted_support_vector_machine'] * 5
weighted_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Weighted Support Vector Machine Model Performance on Train Data: ')
display(weighted_support_vector_machine_performance_train)

Weighted Support Vector Machine Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.964912	weighted_support_vector_machine	train
1	Precision	0.962963	weighted_support_vector_machine	train
2	Recall	0.896552	weighted_support_vector_machine	train
3	F1	0.928571	weighted_support_vector_machine	train
4	AUROC	0.942394	weighted_support_vector_machine	train

##################################
# Evaluating the weighted Support Vector Machine model
# on the test set
##################################
weighted_support_vector_machine_y_hat_test = weighted_support_vector_machine.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_support_vector_machine_performance_test = model_performance_evaluation(y_test, weighted_support_vector_machine_y_hat_test)
weighted_support_vector_machine_performance_test['model'] = ['weighted_support_vector_machine'] * 5
weighted_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Weighted Support Vector Machine Model Performance on Test Data: ')
display(weighted_support_vector_machine_performance_test)

Weighted Support Vector Machine Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.877551	weighted_support_vector_machine	test
1	Precision	0.875000	weighted_support_vector_machine	test
2	Recall	0.583333	weighted_support_vector_machine	test
3	F1	0.700000	weighted_support_vector_machine	test
4	AUROC	0.778153	weighted_support_vector_machine	test

1.3.7 Model Development With SMOTE Upsampling

1.3.7.1 Premodelling Data Description

1. Among the 9 numeric variables determined to have a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable, only 7 were retained with absolute T-Test statistics greater than 5.
   • GDPCAP: T.Test.Statistic=-11.937, T.Test.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, T.Test.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, T.Test.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, T.Test.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, T.Test.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, T.Test.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, T.Test.PValue=0.000
2. Among the 4 categorical predictors determined to have a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable, only 1 was retained with absolute Chi-Square statistics greater than 15.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
3. The SMOTE algorithm from the imblearn.over_sampling Python library API was implemented. The extended model training data by upsampling the minority HIGH CANRAT category applying SMOTE was used.

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 9)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP        float64
LIFEXP        float64
TUBINC        float64
DTHCMD        float64
CO2EMI        float64
GDPCAP        float64
EPISCO        float64
CANRAT       category
HDICAT_VH       uint8
dtype: object

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling,
             kind='reg',
             hue='CANRAT');
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 8)

##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)

0    0.745614
1    0.254386
dtype: float64

##################################
# Initiating an oversampling instance
# on the train data using
# Synthetic Minority Oversampling Technique
##################################
smote = SMOTE(random_state = 88888888)
X_train_smote, y_train_smote = smote.fit_resample(X_train,y_train)

##################################
# Performing a general exploration of the overampled train dataset
##################################
print('Dataset Dimensions: ')
display(X_train_smote.shape)

Dataset Dimensions: 

(170, 8)

##################################
# Validating the class distribution of the overampled train dataset
##################################
y_train_smote.value_counts(normalize = True)

0    0.5
1    0.5
dtype: float64

##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)

Dataset Dimensions: 

(49, 8)

##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)

0    0.755102
1    0.244898
dtype: float64

##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
    metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
    metric_value = [accuracy_score(y_true, y_pred),
                   precision_score(y_true, y_pred),
                   recall_score(y_true, y_pred),
                   f1_score(y_true, y_pred),
                   roc_auc_score(y_true, y_pred)]    
    metric_summary = pd.DataFrame(zip(metric_name, metric_value),
                                  columns=['metric_name','metric_value']) 
    return(metric_summary)

1.3.7.2 Logistic Regression

1. The logistic regression model from the sklearn.linear_model Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • penalty = penalty norm made to vary between L1 and L2
   • solver = algorithm used in the optimization problem made to vary between Saga and Liblinear
   • class_weight = weights associated with classes held constant at a value of None
   • max_iter = maximum number of iterations taken for the solvers to converge held constant at a value of 500
3. The extended model training data by upsampling the minority HIGH CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • penalty = L1 norm
   • solver = Saga
   • class_weight = None
   • max_iter = 500
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9649
   • Precision = 0.9032
   • Recall = 0.9655
   • F1 Score = 0.9333
   • AUROC = 0.9651
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.9183
   • Precision = 0.9000
   • Recall = 0.7500
   • F1 Score = 0.8181
   • AUROC = 0.8614
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'penalty': ['l1', 'l2'],
    'solver': ['liblinear','saga'],
    'class_weight': [None],
    'max_iter': [500],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
upsampled_logistic_regression = GridSearchCV(estimator = logistic_regression, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the upsampled Logistic Regression model
##################################
upsampled_logistic_regression.fit(X_train_smote, y_train_smote)

##################################
# Determining the optimal hyperparameter
# for the Logistic Regression model
##################################
upsampled_logistic_regression.best_score_ 
upsampled_logistic_regression.best_params_

{'C': 1.0,
 'class_weight': None,
 'max_iter': 500,
 'penalty': 'l1',
 'random_state': 88888888,
 'solver': 'saga'}

##################################
# Evaluating the upsampled Logistic Regression model
# on the train set
##################################
upsampled_logistic_regression_y_hat_train = upsampled_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_logistic_regression_performance_train = model_performance_evaluation(y_train, upsampled_logistic_regression_y_hat_train)
upsampled_logistic_regression_performance_train['model'] = ['upsampled_logistic_regression'] * 5
upsampled_logistic_regression_performance_train['set'] = ['train'] * 5
print('Upsampled Logistic Regression Model Performance on Train Data: ')
display(upsampled_logistic_regression_performance_train)

Upsampled Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.964912	upsampled_logistic_regression	train
1	Precision	0.903226	upsampled_logistic_regression	train
2	Recall	0.965517	upsampled_logistic_regression	train
3	F1	0.933333	upsampled_logistic_regression	train
4	AUROC	0.965112	upsampled_logistic_regression	train

##################################
# Evaluating the upsampled Logistic Regression model
# on the test set
##################################
upsampled_logistic_regression_y_hat_test = upsampled_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_logistic_regression_performance_test = model_performance_evaluation(y_test, upsampled_logistic_regression_y_hat_test)
upsampled_logistic_regression_performance_test['model'] = ['upsampled_logistic_regression'] * 5
upsampled_logistic_regression_performance_test['set'] = ['test'] * 5
print('Upsampled Logistic Regression Model Performance on Test Data: ')
display(upsampled_logistic_regression_performance_test)

Upsampled Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.918367	upsampled_logistic_regression	test
1	Precision	0.900000	upsampled_logistic_regression	test
2	Recall	0.750000	upsampled_logistic_regression	test
3	F1	0.818182	upsampled_logistic_regression	test
4	AUROC	0.861486	upsampled_logistic_regression	test

1.3.7.3 Decision Trees

1. The decision tree model from the sklearn.tree Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • class_weight = weights associated with classes held constant at a value of None
3. The extended model training data by upsampling the minority HIGH CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Entropy
   • max_depth = 3
   • min_samples_leaf = 5
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9210
   • Precision = 0.7631
   • Recall = 1.0000
   • F1 Score = 0.8656
   • AUROC = 0.9470
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.7692
   • Recall = 0.8333
   • F1 Score = 0.8000
   • AUROC = 0.8761
7. Considerable difference in the apparent and independent test model performance observed, indicative of the presence of moderate model overfitting.

##################################
# Creating an instance of the 
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
upsampled_decision_tree = GridSearchCV(estimator = decision_tree, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the upsampled Decision Tree model
##################################
upsampled_decision_tree.fit(X_train_smote, y_train_smote)

##################################
# Determining the optimal hyperparameter
# for the Decision Tree model
##################################
upsampled_decision_tree.best_score_ 
upsampled_decision_tree.best_params_

{'class_weight': None,
 'criterion': 'entropy',
 'max_depth': 3,
 'min_samples_leaf': 5,
 'random_state': 88888888}

##################################
# Evaluating the upsampled Decision Tree model
# on the train set
##################################
upsampled_decision_tree_y_hat_train = upsampled_decision_tree.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_decision_tree_performance_train = model_performance_evaluation(y_train, upsampled_decision_tree_y_hat_train)
upsampled_decision_tree_performance_train['model'] = ['upsampled_decision_tree'] * 5
upsampled_decision_tree_performance_train['set'] = ['train'] * 5
print('Upsampled Decision Tree Model Performance on Train Data: ')
display(upsampled_decision_tree_performance_train)

Upsampled Decision Tree Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.921053	upsampled_decision_tree	train
1	Precision	0.763158	upsampled_decision_tree	train
2	Recall	1.000000	upsampled_decision_tree	train
3	F1	0.865672	upsampled_decision_tree	train
4	AUROC	0.947059	upsampled_decision_tree	train

##################################
# Evaluating the upsampled Decision Tree model
# on the test set
##################################
upsampled_decision_tree_y_hat_test = upsampled_decision_tree.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_decision_tree_performance_test = model_performance_evaluation(y_test, upsampled_decision_tree_y_hat_test)
upsampled_decision_tree_performance_test['model'] = ['upsampled_decision_tree'] * 5
upsampled_decision_tree_performance_test['set'] = ['test'] * 5
print('Upsampled Decision Tree Model Performance on Test Data: ')
display(upsampled_decision_tree_performance_test)

Upsampled Decision Tree Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	upsampled_decision_tree	test
1	Precision	0.769231	upsampled_decision_tree	test
2	Recall	0.833333	upsampled_decision_tree	test
3	F1	0.800000	upsampled_decision_tree	test
4	AUROC	0.876126	upsampled_decision_tree	test

1.3.7.4 Random Forest

1. The random forest model from the sklearn.ensemble Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • n_estimators = number of trees in the forest made to vary between 100, 150 and 200
   • max_features = number of features to consider when looking for the best split made to vary between Sqrt and Log2 of n_estimators
   • class_weight = weights associated with classes held constant at a value of None
3. The extended model training data by upsampling the minority HIGH CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Entropy
   • max_depth = 7
   • min_samples_leaf = 3
   • n_estimators = 100
   • max_features = Sqrt n_estimators
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9912
   • Precision = 0.9666
   • Recall = 1.0000
   • F1 Score = 0.9830
   • AUROC = 0.9941
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.9183
   • Precision = 0.9000
   • Recall = 0.7500
   • F1 Score = 0.8181
   • AUROC = 0.8614
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Random Forest model
##################################
random_forest = RandomForestClassifier()

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'n_estimators': [100,150,200],
    'max_features':['sqrt', 'log2'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
upsampled_random_forest = GridSearchCV(estimator = random_forest, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the upsampled Random Forest model
##################################
upsampled_random_forest.fit(X_train_smote, y_train_smote)

##################################
# Determining the optimal hyperparameter
# for the Random Forest model
##################################
upsampled_random_forest.best_score_ 
upsampled_random_forest.best_params_

{'class_weight': None,
 'criterion': 'entropy',
 'max_depth': 7,
 'max_features': 'sqrt',
 'min_samples_leaf': 3,
 'n_estimators': 100,
 'random_state': 88888888}

##################################
# Evaluating the upsampled Random Forest model
# on the train set
##################################
upsampled_random_forest_y_hat_train = upsampled_random_forest.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_random_forest_performance_train = model_performance_evaluation(y_train, upsampled_random_forest_y_hat_train)
upsampled_random_forest_performance_train['model'] = ['upsampled_random_forest'] * 5
upsampled_random_forest_performance_train['set'] = ['train'] * 5
print('Upsampled Random Forest Model Performance on Train Data: ')
display(upsampled_random_forest_performance_train)

Upsampled Random Forest Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.991228	upsampled_random_forest	train
1	Precision	0.966667	upsampled_random_forest	train
2	Recall	1.000000	upsampled_random_forest	train
3	F1	0.983051	upsampled_random_forest	train
4	AUROC	0.994118	upsampled_random_forest	train

##################################
# Evaluating the upsampled Random Forest model
# on the test set
##################################
upsampled_random_forest_y_hat_test = upsampled_random_forest.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_random_forest_performance_test = model_performance_evaluation(y_test, upsampled_random_forest_y_hat_test)
upsampled_random_forest_performance_test['model'] = ['upsampled_random_forest'] * 5
upsampled_random_forest_performance_test['set'] = ['test'] * 5
print('Upsampled Random Forest Model Performance on Test Data: ')
display(upsampled_random_forest_performance_test)

Upsampled Random Forest Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.918367	upsampled_random_forest	test
1	Precision	0.900000	upsampled_random_forest	test
2	Recall	0.750000	upsampled_random_forest	test
3	F1	0.818182	upsampled_random_forest	test
4	AUROC	0.861486	upsampled_random_forest	test

1.3.7.5 Support Vector Machine

1. The support vector machine model from the sklearn.svm Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • kernel = kernel type to be used in the algorithm made to vary between Linear, Poly, RBF and Sigmoid
   • class_weight = weights associated with classes held constant at a value of None
3. The extended model training data by upsampling the minority HIGH CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • kernel = Linear
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9736
   • Precision = 0.9062
   • Recall = 1.0000
   • F1 Score = 0.9508
   • AUROC = 0.9823
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.8181
   • Recall = 0.7500
   • F1 Score = 0.7826
   • AUROC = 0.8479
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Support Vector Machine model
##################################
support_vector_machine = SVC()

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
upsampled_support_vector_machine = GridSearchCV(estimator = support_vector_machine, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the upsampled Support Vector Machine model
##################################
upsampled_support_vector_machine.fit(X_train_smote, y_train_smote)

##################################
# Determining the optimal hyperparameter
# for the Support Vector Machine model
##################################
upsampled_support_vector_machine.best_score_ 
upsampled_support_vector_machine.best_params_

{'C': 1.0, 'class_weight': None, 'kernel': 'linear', 'random_state': 88888888}

##################################
# Evaluating the upsampled Support Vector Machine model
# on the train set
##################################
upsampled_support_vector_machine_y_hat_train = upsampled_support_vector_machine.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_support_vector_machine_performance_train = model_performance_evaluation(y_train, upsampled_support_vector_machine_y_hat_train)
upsampled_support_vector_machine_performance_train['model'] = ['upsampled_support_vector_machine'] * 5
upsampled_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Upsampled Support Vector Machine Model Performance on Train Data: ')
display(upsampled_support_vector_machine_performance_train)

Upsampled Support Vector Machine Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.973684	upsampled_support_vector_machine	train
1	Precision	0.906250	upsampled_support_vector_machine	train
2	Recall	1.000000	upsampled_support_vector_machine	train
3	F1	0.950820	upsampled_support_vector_machine	train
4	AUROC	0.982353	upsampled_support_vector_machine	train

##################################
# Evaluating the upsampled Support Vector Machine model
# on the test set
##################################
upsampled_support_vector_machine_y_hat_test = upsampled_support_vector_machine.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
upsampled_support_vector_machine_performance_test = model_performance_evaluation(y_test, upsampled_support_vector_machine_y_hat_test)
upsampled_support_vector_machine_performance_test['model'] = ['upsampled_support_vector_machine'] * 5
upsampled_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Upsampled Support Vector Machine Model Performance on Test Data: ')
display(upsampled_support_vector_machine_performance_test)

Upsampled Support Vector Machine Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	upsampled_support_vector_machine	test
1	Precision	0.818182	upsampled_support_vector_machine	test
2	Recall	0.750000	upsampled_support_vector_machine	test
3	F1	0.782609	upsampled_support_vector_machine	test
4	AUROC	0.847973	upsampled_support_vector_machine	test

1.3.8 Model Development With CNN Downsampling

1.3.8.1 Premodelling Data Description

1. Among the 9 numeric variables determined to have a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable, only 7 were retained with absolute T-Test statistics greater than 5.
   • GDPCAP: T.Test.Statistic=-11.937, T.Test.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, T.Test.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, T.Test.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, T.Test.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, T.Test.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, T.Test.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, T.Test.PValue=0.000
2. Among the 4 categorical predictors determined to have a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable, only 1 was retained with absolute Chi-Square statistics greater than 15.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
3. The CNN algorithm from the imblearn.under_sampling Python library API was implemented. The reduced model training data by downsampling the majority LOW CANRAT category applying CNN was used.

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 9)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP        float64
LIFEXP        float64
TUBINC        float64
DTHCMD        float64
CO2EMI        float64
GDPCAP        float64
EPISCO        float64
CANRAT       category
HDICAT_VH       uint8
dtype: object

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling,
             kind='reg',
             hue='CANRAT');
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 8)

##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)

0    0.745614
1    0.254386
dtype: float64

##################################
# Initiating an oversampling instance
# on the train data using
# Condense Nearest Neighbors
##################################
cnn = CondensedNearestNeighbour(random_state = 88888888, n_neighbors=3)
X_train_cnn, y_train_cnn = cnn.fit_resample(X_train,y_train)

##################################
# Performing a general exploration of the overampled train dataset
##################################
print('Dataset Dimensions: ')
display(X_train_cnn.shape)

Dataset Dimensions: 

(50, 8)

##################################
# Validating the class distribution of the overampled train dataset
##################################
y_train_cnn.value_counts(normalize = True)

1    0.58
0    0.42
dtype: float64

##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)

Dataset Dimensions: 

(49, 8)

##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)

0    0.755102
1    0.244898
dtype: float64

##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
    metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
    metric_value = [accuracy_score(y_true, y_pred),
                   precision_score(y_true, y_pred),
                   recall_score(y_true, y_pred),
                   f1_score(y_true, y_pred),
                   roc_auc_score(y_true, y_pred)]    
    metric_summary = pd.DataFrame(zip(metric_name, metric_value),
                                  columns=['metric_name','metric_value']) 
    return(metric_summary)

1.3.8.2 Logistic Regression

1. The logistic regression model from the sklearn.linear_model Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • penalty = penalty norm made to vary between L1 and L2
   • solver = algorithm used in the optimization problem made to vary between Saga and Liblinear
   • class_weight = weights associated with classes held constant at a value of None
   • max_iter = maximum number of iterations taken for the solvers to converge held constant at a value of 500
3. The reduced model training data by downsampling the majority LOW CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • penalty = L1 norm
   • solver = Liblinear
   • class_weight = None
   • max_iter = 500
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9473
   • Precision = 0.8484
   • Recall = 0.9655
   • F1 Score = 0.9032
   • AUROC = 0.9533
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.9183
   • Precision = 0.9000
   • Recall = 0.7500
   • F1 Score = 0.8181
   • AUROC = 0.8614
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'penalty': ['l1', 'l2'],
    'solver': ['liblinear','saga'],
    'class_weight': [None],
    'max_iter': [500],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
downsampled_logistic_regression = GridSearchCV(estimator = logistic_regression, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the downsampled Logistic Regression model
##################################
downsampled_logistic_regression.fit(X_train_cnn, y_train_cnn)

##################################
# Determining the optimal hyperparameter
# for the Logistic Regression model
##################################
downsampled_logistic_regression.best_score_ 
downsampled_logistic_regression.best_params_

{'C': 1.0,
 'class_weight': None,
 'max_iter': 500,
 'penalty': 'l1',
 'random_state': 88888888,
 'solver': 'liblinear'}

##################################
# Evaluating the downsampled Logistic Regression model
# on the train set
##################################
downsampled_logistic_regression_y_hat_train = downsampled_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_logistic_regression_performance_train = model_performance_evaluation(y_train, downsampled_logistic_regression_y_hat_train)
downsampled_logistic_regression_performance_train['model'] = ['downsampled_logistic_regression'] * 5
downsampled_logistic_regression_performance_train['set'] = ['train'] * 5
print('Downsampled Logistic Regression Model Performance on Train Data: ')
display(downsampled_logistic_regression_performance_train)

Downsampled Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	downsampled_logistic_regression	train
1	Precision	0.848485	downsampled_logistic_regression	train
2	Recall	0.965517	downsampled_logistic_regression	train
3	F1	0.903226	downsampled_logistic_regression	train
4	AUROC	0.953347	downsampled_logistic_regression	train

##################################
# Evaluating the downsampled Logistic Regression model
# on the test set
##################################
downsampled_logistic_regression_y_hat_test = downsampled_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_logistic_regression_performance_test = model_performance_evaluation(y_test, downsampled_logistic_regression_y_hat_test)
downsampled_logistic_regression_performance_test['model'] = ['downsampled_logistic_regression'] * 5
downsampled_logistic_regression_performance_test['set'] = ['test'] * 5
print('Downsampled Logistic Regression Model Performance on Test Data: ')
display(downsampled_logistic_regression_performance_test)

Downsampled Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.918367	downsampled_logistic_regression	test
1	Precision	0.900000	downsampled_logistic_regression	test
2	Recall	0.750000	downsampled_logistic_regression	test
3	F1	0.818182	downsampled_logistic_regression	test
4	AUROC	0.861486	downsampled_logistic_regression	test

1.3.8.3 Decision Trees

1. The decision tree model from the sklearn.tree Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • class_weight = weights associated with classes held constant at a value of None
3. The reduced model training data by downsampling the majority LOW CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Gini
   • max_depth = 3
   • min_samples_leaf = 5
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9385
   • Precision = 0.9230
   • Recall = 0.8275
   • F1 Score = 0.8727
   • AUROC = 0.9020
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.8888
   • Recall = 0.6666
   • F1 Score = 0.7619
   • AUROC = 0.8198
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
downsampled_decision_tree = GridSearchCV(estimator = decision_tree, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the downsampled Decision Tree model
##################################
downsampled_decision_tree.fit(X_train_cnn, y_train_cnn)

##################################
# Determining the optimal hyperparameter
# for the Decision Tree model
##################################
downsampled_decision_tree.best_score_ 
downsampled_decision_tree.best_params_

{'class_weight': None,
 'criterion': 'gini',
 'max_depth': 3,
 'min_samples_leaf': 5,
 'random_state': 88888888}

##################################
# Evaluating the downsampled Decision Tree model
# on the train set
##################################
downsampled_decision_tree_y_hat_train = downsampled_decision_tree.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_decision_tree_performance_train = model_performance_evaluation(y_train, downsampled_decision_tree_y_hat_train)
downsampled_decision_tree_performance_train['model'] = ['downsampled_decision_tree'] * 5
downsampled_decision_tree_performance_train['set'] = ['train'] * 5
print('Downsampled Decision Tree Model Performance on Train Data: ')
display(downsampled_decision_tree_performance_train)

Downsampled Decision Tree Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.938596	downsampled_decision_tree	train
1	Precision	0.923077	downsampled_decision_tree	train
2	Recall	0.827586	downsampled_decision_tree	train
3	F1	0.872727	downsampled_decision_tree	train
4	AUROC	0.902028	downsampled_decision_tree	train

##################################
# Evaluating the downsampled Decision Tree model
# on the test set
##################################
downsampled_decision_tree_y_hat_test = downsampled_decision_tree.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_decision_tree_performance_test = model_performance_evaluation(y_test, downsampled_decision_tree_y_hat_test)
downsampled_decision_tree_performance_test['model'] = ['downsampled_decision_tree'] * 5
downsampled_decision_tree_performance_test['set'] = ['test'] * 5
print('Downsampled Decision Tree Model Performance on Test Data: ')
display(downsampled_decision_tree_performance_test)

Downsampled Decision Tree Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	downsampled_decision_tree	test
1	Precision	0.888889	downsampled_decision_tree	test
2	Recall	0.666667	downsampled_decision_tree	test
3	F1	0.761905	downsampled_decision_tree	test
4	AUROC	0.819820	downsampled_decision_tree	test

1.3.8.4 Random Forest

1. The random forest model from the sklearn.ensemble Python library API was implemented.
2. The model contains 5 hyperparameters:
   • criterion = function to measure the quality of a split made to vary between Gini, Entropy and Log-Loss
   • max_depth = maximum depth of the tree made to vary between 3, 5 and 7
   • min_samples_leaf = minimum number of samples required to split an internal node made to vary between 3, 5 and 10
   • n_estimators = number of trees in the forest made to vary between 100, 150 and 200
   • max_features = number of features to consider when looking for the best split made to vary between Sqrt and Log2 of n_estimators
   • class_weight = weights associated with classes held constant at a value of None
3. The reduced model training data by downsampling the majority LOW CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • criterion = Gini
   • max_depth = 3
   • min_samples_leaf = 3
   • n_estimators = 100
   • max_features = Sqrt n_estimators
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9649
   • Precision = 0.9032
   • Recall = 0.9655
   • F1 Score = 0.9333
   • AUROC = 0.9651
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.8888
   • Recall = 0.6666
   • F1 Score = 0.7619
   • AUROC = 0.8198
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Random Forest model
##################################
random_forest = RandomForestClassifier()

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'n_estimators': [100,150,200],
    'max_features':['sqrt', 'log2'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
downsampled_random_forest = GridSearchCV(estimator = random_forest, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the downsampled Random Forest model
##################################
downsampled_random_forest.fit(X_train_cnn, y_train_cnn)

##################################
# Determining the optimal hyperparameter
# for the Random Forest model
##################################
downsampled_random_forest.best_score_ 
downsampled_random_forest.best_params_

{'class_weight': None,
 'criterion': 'gini',
 'max_depth': 3,
 'max_features': 'sqrt',
 'min_samples_leaf': 3,
 'n_estimators': 100,
 'random_state': 88888888}

##################################
# Evaluating the downsampled Random Forest model
# on the train set
##################################
downsampled_random_forest_y_hat_train = downsampled_random_forest.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_random_forest_performance_train = model_performance_evaluation(y_train, downsampled_random_forest_y_hat_train)
downsampled_random_forest_performance_train['model'] = ['downsampled_random_forest'] * 5
downsampled_random_forest_performance_train['set'] = ['train'] * 5
print('Downsampled Random Forest Model Performance on Train Data: ')
display(downsampled_random_forest_performance_train)

Downsampled Random Forest Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.964912	downsampled_random_forest	train
1	Precision	0.903226	downsampled_random_forest	train
2	Recall	0.965517	downsampled_random_forest	train
3	F1	0.933333	downsampled_random_forest	train
4	AUROC	0.965112	downsampled_random_forest	train

##################################
# Evaluating the downsampled Random Forest model
# on the test set
##################################
downsampled_random_forest_y_hat_test = downsampled_random_forest.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_random_forest_performance_test = model_performance_evaluation(y_test, downsampled_random_forest_y_hat_test)
downsampled_random_forest_performance_test['model'] = ['downsampled_random_forest'] * 5
downsampled_random_forest_performance_test['set'] = ['test'] * 5
print('Downsampled Random Forest Model Performance on Test Data: ')
display(downsampled_random_forest_performance_test)

Downsampled Random Forest Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	downsampled_random_forest	test
1	Precision	0.888889	downsampled_random_forest	test
2	Recall	0.666667	downsampled_random_forest	test
3	F1	0.761905	downsampled_random_forest	test
4	AUROC	0.819820	downsampled_random_forest	test

1.3.8.5 Support Vector Machine

1. The support vector machine model from the sklearn.svm Python library API was implemented.
2. The model contains 5 hyperparameters:
   • C = inverse of regularization strength held constant at a value of 1
   • kernel = kernel type to be used in the algorithm made to vary between Linear, Poly, RBF and Sigmoid
   • class_weight = weights associated with classes held constant at a value of None
3. The reduced model training data by downsampling the majority LOW CANRAT category was used.
4. Hyperparameter tuning was conducted using the 5-fold cross-validation method with optimal model performance using the F1 score determined for:
   • C = 1
   • kernel = Linear
   • class_weight = None
5. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9561
   • Precision = 0.9285
   • Recall = 0.8965
   • F1 Score = 0.9122
   • AUROC = 0.9365
6. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.8979
   • Precision = 0.8888
   • Recall = 0.6666
   • F1 Score = 0.7619
   • AUROC = 0.8198
7. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Creating an instance of the 
# Support Vector Machine model
##################################
support_vector_machine = SVC()

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
downsampled_support_vector_machine = GridSearchCV(estimator = support_vector_machine, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the downsampled Support Vector Machine model
##################################
downsampled_support_vector_machine.fit(X_train_cnn, y_train_cnn)

##################################
# Determining the optimal hyperparameter
# for the Support Vector Machine model
##################################
downsampled_support_vector_machine.best_score_ 
downsampled_support_vector_machine.best_params_

{'C': 1.0, 'class_weight': None, 'kernel': 'linear', 'random_state': 88888888}

##################################
# Evaluating the downsampled Support Vector Machine model
# on the train set
##################################
downsampled_support_vector_machine_y_hat_train = downsampled_support_vector_machine.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_support_vector_machine_performance_train = model_performance_evaluation(y_train, downsampled_support_vector_machine_y_hat_train)
downsampled_support_vector_machine_performance_train['model'] = ['downsampled_support_vector_machine'] * 5
downsampled_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Downsampled Support Vector Machine Model Performance on Train Data: ')
display(downsampled_support_vector_machine_performance_train)

Downsampled Support Vector Machine Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.956140	downsampled_support_vector_machine	train
1	Precision	0.928571	downsampled_support_vector_machine	train
2	Recall	0.896552	downsampled_support_vector_machine	train
3	F1	0.912281	downsampled_support_vector_machine	train
4	AUROC	0.936511	downsampled_support_vector_machine	train

##################################
# Evaluating the downsampled Support Vector Machine model
# on the test set
##################################
downsampled_support_vector_machine_y_hat_test = downsampled_support_vector_machine.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
downsampled_support_vector_machine_performance_test = model_performance_evaluation(y_test, downsampled_support_vector_machine_y_hat_test)
downsampled_support_vector_machine_performance_test['model'] = ['downsampled_support_vector_machine'] * 5
downsampled_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Downsampled Support Vector Machine Model Performance on Test Data: ')
display(downsampled_support_vector_machine_performance_test)

Downsampled Support Vector Machine Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	downsampled_support_vector_machine	test
1	Precision	0.888889	downsampled_support_vector_machine	test
2	Recall	0.666667	downsampled_support_vector_machine	test
3	F1	0.761905	downsampled_support_vector_machine	test
4	AUROC	0.819820	downsampled_support_vector_machine	test

1.3.9 Model Development With Stacking Ensemble Learning

1.3.9.1 Premodelling Data Description

1. Among the formulated versions of the logistic regression model, the model which applied class weights demonstrated the best independent test model performance. Considerable difference in the apparent and independent test model performance was observed, indicative of the presence of moderate model overfitting.
   • Accuracy = 0.9387
   • Precision = 0.8461
   • Recall = 0.9167
   • F1 Score = 0.8800
   • AUROC = 0.9313
2. Among the formulated versions of the decision tree model, the model which applied upsampling of the minority class using SMOTE demonstrated the best independent test model performance. Considerable difference in the apparent and independent test model performance was observed, indicative of the presence of moderate model overfitting.
   • Accuracy = 0.8979
   • Precision = 0.7692
   • Recall = 0.8333
   • F1 Score = 0.8000
   • AUROC = 0.8761
3. Among the formulated versions of the random forest model, the model which applied upsampling of the minority class using SMOTE demonstrated the best independent test model performance. High difference in the apparent and independent test model performance was observed, indicative of the presence of excessive model overfitting.
   • Accuracy = 0.9387
   • Precision = 0.8461
   • Recall = 0.9167
   • F1 Score = 0.8800
   • AUROC = 0.9313
4. Among the formulated versions of the support vector machine model, the model which applied upsampling of the minority class using SMOTE demonstrated the best independent test model performance. High difference in the apparent and independent test model performance was observed, indicative of the presence of excessive model overfitting.
   • Accuracy = 0.8979
   • Precision = 0.8181
   • Recall = 0.7500
   • F1 Score = 0.7826
   • AUROC = 0.8479
5. All individual formulated models which applied upsampling of the minority class using SMOTE were used to generate the base-learners for the stacking algorithm.

##################################
# Consolidating all the
# Logistic Regression
# model performance measures
##################################
logistic_regression_performance_comparison = pd.concat([optimal_logistic_regression_performance_train, 
                                                        optimal_logistic_regression_performance_test,
                                                        weighted_logistic_regression_performance_train, 
                                                        weighted_logistic_regression_performance_test,
                                                        upsampled_logistic_regression_performance_train, 
                                                        upsampled_logistic_regression_performance_test,
                                                        downsampled_logistic_regression_performance_train, 
                                                        downsampled_logistic_regression_performance_test], 
                                                       ignore_index=True)
print('Consolidated Logistic Regression Model Performance on Train and Test Data: ')
display(logistic_regression_performance_comparison)

Consolidated Logistic Regression Model Performance on Train and Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	optimal_logistic_regression	train
1	Precision	0.870968	optimal_logistic_regression	train
2	Recall	0.931034	optimal_logistic_regression	train
3	F1	0.900000	optimal_logistic_regression	train
4	AUROC	0.941988	optimal_logistic_regression	train
5	Accuracy	0.897959	optimal_logistic_regression	test
6	Precision	0.888889	optimal_logistic_regression	test
7	Recall	0.666667	optimal_logistic_regression	test
8	F1	0.761905	optimal_logistic_regression	test
9	AUROC	0.819820	optimal_logistic_regression	test
10	Accuracy	0.894737	weighted_logistic_regression	train
11	Precision	0.707317	weighted_logistic_regression	train
12	Recall	1.000000	weighted_logistic_regression	train
13	F1	0.828571	weighted_logistic_regression	train
14	AUROC	0.929412	weighted_logistic_regression	train
15	Accuracy	0.938776	weighted_logistic_regression	test
16	Precision	0.846154	weighted_logistic_regression	test
17	Recall	0.916667	weighted_logistic_regression	test
18	F1	0.880000	weighted_logistic_regression	test
19	AUROC	0.931306	weighted_logistic_regression	test
20	Accuracy	0.964912	upsampled_logistic_regression	train
21	Precision	0.903226	upsampled_logistic_regression	train
22	Recall	0.965517	upsampled_logistic_regression	train
23	F1	0.933333	upsampled_logistic_regression	train
24	AUROC	0.965112	upsampled_logistic_regression	train
25	Accuracy	0.918367	upsampled_logistic_regression	test
26	Precision	0.900000	upsampled_logistic_regression	test
27	Recall	0.750000	upsampled_logistic_regression	test
28	F1	0.818182	upsampled_logistic_regression	test
29	AUROC	0.861486	upsampled_logistic_regression	test
30	Accuracy	0.947368	downsampled_logistic_regression	train
31	Precision	0.848485	downsampled_logistic_regression	train
32	Recall	0.965517	downsampled_logistic_regression	train
33	F1	0.903226	downsampled_logistic_regression	train
34	AUROC	0.953347	downsampled_logistic_regression	train
35	Accuracy	0.918367	downsampled_logistic_regression	test
36	Precision	0.900000	downsampled_logistic_regression	test
37	Recall	0.750000	downsampled_logistic_regression	test
38	F1	0.818182	downsampled_logistic_regression	test
39	AUROC	0.861486	downsampled_logistic_regression	test

##################################
# Consolidating all the F1 score
# model performance measures
##################################
logistic_regression_performance_comparison_F1 = logistic_regression_performance_comparison[logistic_regression_performance_comparison['metric_name']=='F1']
logistic_regression_performance_comparison_F1_train = logistic_regression_performance_comparison_F1[logistic_regression_performance_comparison_F1['set']=='train'].loc[:,"metric_value"]
logistic_regression_performance_comparison_F1_test = logistic_regression_performance_comparison_F1[logistic_regression_performance_comparison_F1['set']=='test'].loc[:,"metric_value"]

##################################
# Combining all the F1 score
# model performance measures
# between train and test sets
##################################
logistic_regression_performance_comparison_F1_plot = pd.DataFrame({'train': logistic_regression_performance_comparison_F1_train.values,
                                                                   'test': logistic_regression_performance_comparison_F1_test.values},
                                                                  index=logistic_regression_performance_comparison_F1['model'].unique())
logistic_regression_performance_comparison_F1_plot

	train	test
optimal_logistic_regression	0.900000	0.761905
weighted_logistic_regression	0.828571	0.880000
upsampled_logistic_regression	0.933333	0.818182
downsampled_logistic_regression	0.903226	0.818182

##################################
# Plotting all the F1 score
# model performance measures
# between train and test sets
##################################
logistic_regression_performance_comparison_F1_plot = logistic_regression_performance_comparison_F1_plot.plot.barh(figsize=(10, 6))
logistic_regression_performance_comparison_F1_plot.set_xlim(0.00,1.00)
logistic_regression_performance_comparison_F1_plot.set_title("Model Comparison by F1 Score Performance on Test Data")
logistic_regression_performance_comparison_F1_plot.set_xlabel("F1 Score Performance")
logistic_regression_performance_comparison_F1_plot.set_ylabel("Logistic Regression Model")
logistic_regression_performance_comparison_F1_plot.grid(False)
logistic_regression_performance_comparison_F1_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in logistic_regression_performance_comparison_F1_plot.containers:
    logistic_regression_performance_comparison_F1_plot.bar_label(container, fmt='%.5f', padding=-50)

##################################
# Plotting the confusion matrices
# for all the Logistic Regression models
##################################
classifiers = {"optimal_logistic_regression": optimal_logistic_regression,
               "weighted_logistic_regression": weighted_logistic_regression,
               "upsampled_logistic_regression": upsampled_logistic_regression,
               "downsampled_logistic_regression": downsampled_logistic_regression}

fig, axes = plt.subplots(1, 4, figsize=(20, 5))
for i, (key, classifier) in enumerate(classifiers.items()):
    y_pred = classifier.predict(X_test)
    cf_matrix = confusion_matrix(y_test, y_pred)
    disp = ConfusionMatrixDisplay(cf_matrix)
    disp.plot(ax=axes[i], xticks_rotation=0)
    disp.ax_.grid(False)
    disp.ax_.set_title(key)
    disp.im_.colorbar.remove()

fig.colorbar(disp.im_, ax=axes)
plt.show()

##################################
# Consolidating all the
# Decision Tree
# model performance measures
##################################
decision_tree_performance_comparison = pd.concat([optimal_decision_tree_performance_train, 
                                                  optimal_decision_tree_performance_test,
                                                  weighted_decision_tree_performance_train, 
                                                  weighted_decision_tree_performance_test,
                                                  upsampled_decision_tree_performance_train, 
                                                  upsampled_decision_tree_performance_test,
                                                  downsampled_decision_tree_performance_train, 
                                                  downsampled_decision_tree_performance_test], 
                                                 ignore_index=True)
print('Consolidated Decision Tree Model Performance on Train and Test Data: ')
display(decision_tree_performance_comparison)

Consolidated Decision Tree Model Performance on Train and Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.973684	optimal_decision_tree	train
1	Precision	1.000000	optimal_decision_tree	train
2	Recall	0.896552	optimal_decision_tree	train
3	F1	0.945455	optimal_decision_tree	train
4	AUROC	0.948276	optimal_decision_tree	train
5	Accuracy	0.857143	optimal_decision_tree	test
6	Precision	0.857143	optimal_decision_tree	test
7	Recall	0.500000	optimal_decision_tree	test
8	F1	0.631579	optimal_decision_tree	test
9	AUROC	0.736486	optimal_decision_tree	test
10	Accuracy	0.956140	weighted_decision_tree	train
11	Precision	0.852941	weighted_decision_tree	train
12	Recall	1.000000	weighted_decision_tree	train
13	F1	0.920635	weighted_decision_tree	train
14	AUROC	0.970588	weighted_decision_tree	train
15	Accuracy	0.897959	weighted_decision_tree	test
16	Precision	0.769231	weighted_decision_tree	test
17	Recall	0.833333	weighted_decision_tree	test
18	F1	0.800000	weighted_decision_tree	test
19	AUROC	0.876126	weighted_decision_tree	test
20	Accuracy	0.921053	upsampled_decision_tree	train
21	Precision	0.763158	upsampled_decision_tree	train
22	Recall	1.000000	upsampled_decision_tree	train
23	F1	0.865672	upsampled_decision_tree	train
24	AUROC	0.947059	upsampled_decision_tree	train
25	Accuracy	0.897959	upsampled_decision_tree	test
26	Precision	0.769231	upsampled_decision_tree	test
27	Recall	0.833333	upsampled_decision_tree	test
28	F1	0.800000	upsampled_decision_tree	test
29	AUROC	0.876126	upsampled_decision_tree	test
30	Accuracy	0.938596	downsampled_decision_tree	train
31	Precision	0.923077	downsampled_decision_tree	train
32	Recall	0.827586	downsampled_decision_tree	train
33	F1	0.872727	downsampled_decision_tree	train
34	AUROC	0.902028	downsampled_decision_tree	train
35	Accuracy	0.897959	downsampled_decision_tree	test
36	Precision	0.888889	downsampled_decision_tree	test
37	Recall	0.666667	downsampled_decision_tree	test
38	F1	0.761905	downsampled_decision_tree	test
39	AUROC	0.819820	downsampled_decision_tree	test

##################################
# Consolidating all the F1 score
# model performance measures
##################################
decision_tree_performance_comparison_F1 = decision_tree_performance_comparison[decision_tree_performance_comparison['metric_name']=='F1']
decision_tree_performance_comparison_F1_train = decision_tree_performance_comparison_F1[decision_tree_performance_comparison_F1['set']=='train'].loc[:,"metric_value"]
decision_tree_performance_comparison_F1_test = decision_tree_performance_comparison_F1[decision_tree_performance_comparison_F1['set']=='test'].loc[:,"metric_value"]

##################################
# Combining all the F1 score
# model performance measures
# between train and test sets
##################################
decision_tree_performance_comparison_F1_plot = pd.DataFrame({'train': decision_tree_performance_comparison_F1_train.values,
                                                             'test': decision_tree_performance_comparison_F1_test.values},
                                                            index=decision_tree_performance_comparison_F1['model'].unique())
decision_tree_performance_comparison_F1_plot

	train	test
optimal_decision_tree	0.945455	0.631579
weighted_decision_tree	0.920635	0.800000
upsampled_decision_tree	0.865672	0.800000
downsampled_decision_tree	0.872727	0.761905

##################################
# Plotting all the F1 score
# model performance measures
# between train and test sets
##################################
decision_tree_performance_comparison_F1_plot = decision_tree_performance_comparison_F1_plot.plot.barh(figsize=(10, 6))
decision_tree_performance_comparison_F1_plot.set_xlim(0.00,1.00)
decision_tree_performance_comparison_F1_plot.set_title("Model Comparison by F1 Score Performance on Test Data")
decision_tree_performance_comparison_F1_plot.set_xlabel("F1 Score Performance")
decision_tree_performance_comparison_F1_plot.set_ylabel("Decision Tree Model")
decision_tree_performance_comparison_F1_plot.grid(False)
decision_tree_performance_comparison_F1_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in decision_tree_performance_comparison_F1_plot.containers:
    decision_tree_performance_comparison_F1_plot.bar_label(container, fmt='%.5f', padding=-50)

##################################
# Plotting the confusion matrices
# for all the Decision Tree models
##################################
classifiers = {"optimal_decision_tree": optimal_decision_tree,
               "weighted_decision_tree": weighted_decision_tree,
               "upsampled_decision_tree": upsampled_decision_tree,
               "downsampled_decision_tree": downsampled_decision_tree}

fig, axes = plt.subplots(1, 4, figsize=(20, 5))
for i, (key, classifier) in enumerate(classifiers.items()):
    y_pred = classifier.predict(X_test)
    cf_matrix = confusion_matrix(y_test, y_pred)
    disp = ConfusionMatrixDisplay(cf_matrix)
    disp.plot(ax=axes[i], xticks_rotation=0)
    disp.ax_.grid(False)
    disp.ax_.set_title(key)
    disp.im_.colorbar.remove()

fig.colorbar(disp.im_, ax=axes)
plt.show()

##################################
# Consolidating all the
# Random Forest
# model performance measures
##################################
random_forest_performance_comparison = pd.concat([optimal_random_forest_performance_train, 
                                                  optimal_random_forest_performance_test,
                                                  weighted_random_forest_performance_train, 
                                                  weighted_random_forest_performance_test,
                                                  upsampled_random_forest_performance_train, 
                                                  upsampled_random_forest_performance_test,
                                                  downsampled_random_forest_performance_train, 
                                                  downsampled_random_forest_performance_test], 
                                                 ignore_index=True)
print('Consolidated Random Forest Model Performance on Train and Test Data: ')
display(random_forest_performance_comparison)

Consolidated Random Forest Model Performance on Train and Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.956140	optimal_random_forest	train
1	Precision	0.928571	optimal_random_forest	train
2	Recall	0.896552	optimal_random_forest	train
3	F1	0.912281	optimal_random_forest	train
4	AUROC	0.936511	optimal_random_forest	train
5	Accuracy	0.877551	optimal_random_forest	test
6	Precision	0.875000	optimal_random_forest	test
7	Recall	0.583333	optimal_random_forest	test
8	F1	0.700000	optimal_random_forest	test
9	AUROC	0.778153	optimal_random_forest	test
10	Accuracy	0.973684	weighted_random_forest	train
11	Precision	0.906250	weighted_random_forest	train
12	Recall	1.000000	weighted_random_forest	train
13	F1	0.950820	weighted_random_forest	train
14	AUROC	0.982353	weighted_random_forest	train
15	Accuracy	0.897959	weighted_random_forest	test
16	Precision	0.888889	weighted_random_forest	test
17	Recall	0.666667	weighted_random_forest	test
18	F1	0.761905	weighted_random_forest	test
19	AUROC	0.819820	weighted_random_forest	test
20	Accuracy	0.991228	upsampled_random_forest	train
21	Precision	0.966667	upsampled_random_forest	train
22	Recall	1.000000	upsampled_random_forest	train
23	F1	0.983051	upsampled_random_forest	train
24	AUROC	0.994118	upsampled_random_forest	train
25	Accuracy	0.918367	upsampled_random_forest	test
26	Precision	0.900000	upsampled_random_forest	test
27	Recall	0.750000	upsampled_random_forest	test
28	F1	0.818182	upsampled_random_forest	test
29	AUROC	0.861486	upsampled_random_forest	test
30	Accuracy	0.964912	downsampled_random_forest	train
31	Precision	0.903226	downsampled_random_forest	train
32	Recall	0.965517	downsampled_random_forest	train
33	F1	0.933333	downsampled_random_forest	train
34	AUROC	0.965112	downsampled_random_forest	train
35	Accuracy	0.897959	downsampled_random_forest	test
36	Precision	0.888889	downsampled_random_forest	test
37	Recall	0.666667	downsampled_random_forest	test
38	F1	0.761905	downsampled_random_forest	test
39	AUROC	0.819820	downsampled_random_forest	test

##################################
# Consolidating all the F1 score
# model performance measures
##################################
random_forest_performance_comparison_F1 = random_forest_performance_comparison[random_forest_performance_comparison['metric_name']=='F1']
random_forest_performance_comparison_F1_train = random_forest_performance_comparison_F1[random_forest_performance_comparison_F1['set']=='train'].loc[:,"metric_value"]
random_forest_performance_comparison_F1_test = random_forest_performance_comparison_F1[random_forest_performance_comparison_F1['set']=='test'].loc[:,"metric_value"]

##################################
# Combining all the F1 score
# model performance measures
# between train and test sets
##################################
random_forest_performance_comparison_F1_plot = pd.DataFrame({'train': random_forest_performance_comparison_F1_train.values,
                                                             'test': random_forest_performance_comparison_F1_test.values},
                                                            index=random_forest_performance_comparison_F1['model'].unique())
random_forest_performance_comparison_F1_plot

	train	test
optimal_random_forest	0.912281	0.700000
weighted_random_forest	0.950820	0.761905
upsampled_random_forest	0.983051	0.818182
downsampled_random_forest	0.933333	0.761905

##################################
# Plotting all the F1 score
# model performance measures
# between train and test sets
##################################
random_forest_performance_comparison_F1_plot = random_forest_performance_comparison_F1_plot.plot.barh(figsize=(10, 6))
random_forest_performance_comparison_F1_plot.set_xlim(0.00,1.00)
random_forest_performance_comparison_F1_plot.set_title("Model Comparison by F1 Score Performance on Test Data")
random_forest_performance_comparison_F1_plot.set_xlabel("F1 Score Performance")
random_forest_performance_comparison_F1_plot.set_ylabel("Random Forest Model")
random_forest_performance_comparison_F1_plot.grid(False)
random_forest_performance_comparison_F1_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in random_forest_performance_comparison_F1_plot.containers:
    random_forest_performance_comparison_F1_plot.bar_label(container, fmt='%.5f', padding=-50)

##################################
# Plotting the confusion matrices
# for all the Random Forest models
##################################
classifiers = {"optimal_random_forest": optimal_random_forest,
               "weighted_random_forest": weighted_random_forest,
               "upsampled_random_forest": upsampled_random_forest,
               "downsampled_random_forest": downsampled_random_forest}

fig, axes = plt.subplots(1, 4, figsize=(20, 5))
for i, (key, classifier) in enumerate(classifiers.items()):
    y_pred = classifier.predict(X_test)
    cf_matrix = confusion_matrix(y_test, y_pred)
    disp = ConfusionMatrixDisplay(cf_matrix)
    disp.plot(ax=axes[i], xticks_rotation=0)
    disp.ax_.grid(False)
    disp.ax_.set_title(key)
    disp.im_.colorbar.remove()

fig.colorbar(disp.im_, ax=axes)
plt.show()

##################################
# Consolidating all the
# Support Vector Machine
# model performance measures
##################################
support_vector_machine_performance_comparison = pd.concat([optimal_support_vector_machine_performance_train, 
                                                  optimal_support_vector_machine_performance_test,
                                                  weighted_support_vector_machine_performance_train, 
                                                  weighted_support_vector_machine_performance_test,
                                                  upsampled_support_vector_machine_performance_train, 
                                                  upsampled_support_vector_machine_performance_test,
                                                  downsampled_support_vector_machine_performance_train, 
                                                  downsampled_support_vector_machine_performance_test], 
                                                 ignore_index=True)
print('Consolidated Support Vector Machine Model Performance on Train and Test Data: ')
display(support_vector_machine_performance_comparison)

Consolidated Support Vector Machine Model Performance on Train and Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	optimal_support_vector_machine	train
1	Precision	0.960000	optimal_support_vector_machine	train
2	Recall	0.827586	optimal_support_vector_machine	train
3	F1	0.888889	optimal_support_vector_machine	train
4	AUROC	0.907911	optimal_support_vector_machine	train
5	Accuracy	0.857143	optimal_support_vector_machine	test
6	Precision	0.857143	optimal_support_vector_machine	test
7	Recall	0.500000	optimal_support_vector_machine	test
8	F1	0.631579	optimal_support_vector_machine	test
9	AUROC	0.736486	optimal_support_vector_machine	test
10	Accuracy	0.964912	weighted_support_vector_machine	train
11	Precision	0.962963	weighted_support_vector_machine	train
12	Recall	0.896552	weighted_support_vector_machine	train
13	F1	0.928571	weighted_support_vector_machine	train
14	AUROC	0.942394	weighted_support_vector_machine	train
15	Accuracy	0.877551	weighted_support_vector_machine	test
16	Precision	0.875000	weighted_support_vector_machine	test
17	Recall	0.583333	weighted_support_vector_machine	test
18	F1	0.700000	weighted_support_vector_machine	test
19	AUROC	0.778153	weighted_support_vector_machine	test
20	Accuracy	0.973684	upsampled_support_vector_machine	train
21	Precision	0.906250	upsampled_support_vector_machine	train
22	Recall	1.000000	upsampled_support_vector_machine	train
23	F1	0.950820	upsampled_support_vector_machine	train
24	AUROC	0.982353	upsampled_support_vector_machine	train
25	Accuracy	0.897959	upsampled_support_vector_machine	test
26	Precision	0.818182	upsampled_support_vector_machine	test
27	Recall	0.750000	upsampled_support_vector_machine	test
28	F1	0.782609	upsampled_support_vector_machine	test
29	AUROC	0.847973	upsampled_support_vector_machine	test
30	Accuracy	0.956140	downsampled_support_vector_machine	train
31	Precision	0.928571	downsampled_support_vector_machine	train
32	Recall	0.896552	downsampled_support_vector_machine	train
33	F1	0.912281	downsampled_support_vector_machine	train
34	AUROC	0.936511	downsampled_support_vector_machine	train
35	Accuracy	0.897959	downsampled_support_vector_machine	test
36	Precision	0.888889	downsampled_support_vector_machine	test
37	Recall	0.666667	downsampled_support_vector_machine	test
38	F1	0.761905	downsampled_support_vector_machine	test
39	AUROC	0.819820	downsampled_support_vector_machine	test

##################################
# Consolidating all the F1 score
# model performance measures
##################################
support_vector_machine_performance_comparison_F1 = support_vector_machine_performance_comparison[support_vector_machine_performance_comparison['metric_name']=='F1']
support_vector_machine_performance_comparison_F1_train = support_vector_machine_performance_comparison_F1[support_vector_machine_performance_comparison_F1['set']=='train'].loc[:,"metric_value"]
support_vector_machine_performance_comparison_F1_test = support_vector_machine_performance_comparison_F1[support_vector_machine_performance_comparison_F1['set']=='test'].loc[:,"metric_value"]

##################################
# Combining all the F1 score
# model performance measures
# between train and test sets
##################################
support_vector_machine_performance_comparison_F1_plot = pd.DataFrame({'train': support_vector_machine_performance_comparison_F1_train.values,
                                                                      'test': support_vector_machine_performance_comparison_F1_test.values},
                                                                     index=support_vector_machine_performance_comparison_F1['model'].unique())
support_vector_machine_performance_comparison_F1_plot

	train	test
optimal_support_vector_machine	0.888889	0.631579
weighted_support_vector_machine	0.928571	0.700000
upsampled_support_vector_machine	0.950820	0.782609
downsampled_support_vector_machine	0.912281	0.761905

##################################
# Plotting all the F1 score
# model performance measures
# between train and test sets
##################################
support_vector_machine_performance_comparison_F1_plot = support_vector_machine_performance_comparison_F1_plot.plot.barh(figsize=(10, 6))
support_vector_machine_performance_comparison_F1_plot.set_xlim(0.00,1.00)
support_vector_machine_performance_comparison_F1_plot.set_title("Model Comparison by F1 Score Performance on Test Data")
support_vector_machine_performance_comparison_F1_plot.set_xlabel("F1 Score Performance")
support_vector_machine_performance_comparison_F1_plot.set_ylabel("Support Vector Machine Model")
support_vector_machine_performance_comparison_F1_plot.grid(False)
support_vector_machine_performance_comparison_F1_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in support_vector_machine_performance_comparison_F1_plot.containers:
    support_vector_machine_performance_comparison_F1_plot.bar_label(container, fmt='%.5f', padding=-50)

##################################
# Plotting the confusion matrices
# for all the Support Vector Machine models
##################################
classifiers = {"optimal_support_vector_machine": optimal_support_vector_machine,
               "weighted_support_vector_machine": weighted_support_vector_machine,
               "upsampled_support_vector_machine": upsampled_support_vector_machine,
               "downsampled_support_vector_machine": downsampled_support_vector_machine}

fig, axes = plt.subplots(1, 4, figsize=(20, 5))
for i, (key, classifier) in enumerate(classifiers.items()):
    y_pred = classifier.predict(X_test)
    cf_matrix = confusion_matrix(y_test, y_pred)
    disp = ConfusionMatrixDisplay(cf_matrix)
    disp.plot(ax=axes[i], xticks_rotation=0)
    disp.ax_.grid(False)
    disp.ax_.set_title(key)
    disp.im_.colorbar.remove()

fig.colorbar(disp.im_, ax=axes)
plt.show()

1.3.9.2 Logistic Regression

1. The logistic regression model from the sklearn.linear_model Python library API was implemented as a meta-learner for the stacking algorithm.
2. The model used default hyperparameters with no tuning applied:
   • C = inverse of regularization strength held constant at a value of 1
   • penalty = penalty norm held constant at a value of L2
   • solver = algorithm used in the optimization problem held constant at a value of Lbfgs
   • class_weight = weights associated with classes held constant at a value of 25-75 between classes 0 and 1
   • max_iter = maximum number of iterations taken for the solvers to converge held constant at a value of 500
3. The original data reflecting a 3:1 class imbalance between the LOW and HIGH CANRAT categories was used for model training and testing.
4. The apparent model performance of the optimal model is summarized as follows:
   • Accuracy = 0.9736
   • Precision = 0.9062
   • Recall = 1.0000
   • F1 Score = 0.9508
   • AUROC = 0.9823
5. The independent test model performance of the final model is summarized as follows:
   • Accuracy = 0.9183
   • Precision = 0.9000
   • Recall = 0.7500
   • F1 Score = 0.8181
   • AUROC = 0.8614
6. High difference in the apparent and independent test model performance observed, indicative of the presence of excessive model overfitting.

##################################
# Formulating the base learners
# using the optimal hyperparameters
# for the upsampled models
##################################
base_learners = [('LR', LogisticRegression(C=1.0,
                                          class_weight=None,
                                          max_iter=500,
                                          penalty='l1',
                                          random_state=88888888,
                                          solver='saga')),
                ('DT', DecisionTreeClassifier(class_weight=None,
                                              criterion='entropy',
                                              max_depth=3,
                                              min_samples_leaf=5,
                                              random_state=88888888)),
                ('RF', RandomForestClassifier(class_weight=None,
                                              criterion='entropy',
                                              max_depth=7,
                                              max_features='sqrt',
                                              min_samples_leaf=3,
                                              n_estimators=100,
                                              random_state=88888888)),
               ('SVM', SVC(class_weight=None,
                           C=1.0,
                           kernel='linear',
                           random_state=88888888))]

##################################
# Formulating the meta learner
# using default hyperparameters
##################################
meta_learner = LogisticRegression(C=1.0,
                                  class_weight=None,
                                  max_iter=500,
                                  random_state=88888888)

##################################
# Formulating the stacked model
# using the base and meta learners
##################################
stacked_logistic_regression = StackingClassifier(estimators=base_learners, final_estimator=meta_learner)

##################################
# Fitting the meta Logistic Regression model
##################################
stacked_logistic_regression.fit(X_train_smote, y_train_smote)

StackingClassifier(estimators=[('LR',
                                LogisticRegression(max_iter=500, penalty='l1',
                                                   random_state=88888888,
                                                   solver='saga')),
                               ('DT',
                                DecisionTreeClassifier(criterion='entropy',
                                                       max_depth=3,
                                                       min_samples_leaf=5,
                                                       random_state=88888888)),
                               ('RF',
                                RandomForestClassifier(criterion='entropy',
                                                       max_depth=7,
                                                       min_samples_leaf=3,
                                                       random_state=88888888)),
                               ('SVM',
                                SVC(kernel='linear', random_state=88888888))],
                   final_estimator=LogisticRegression(max_iter=500,
                                                      random_state=88888888))

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

##################################
# Evaluating the stacked Logistic Regression model
# on the train set
##################################
stacked_logistic_regression_y_hat_train = stacked_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
stacked_logistic_regression_performance_train = model_performance_evaluation(y_train, stacked_logistic_regression_y_hat_train)
stacked_logistic_regression_performance_train['model'] = ['stacked_logistic_regression'] * 5
stacked_logistic_regression_performance_train['set'] = ['train'] * 5
print('Stacked Logistic Regression Model Performance on Train Data: ')
display(stacked_logistic_regression_performance_train)

Stacked Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.973684	stacked_logistic_regression	train
1	Precision	0.906250	stacked_logistic_regression	train
2	Recall	1.000000	stacked_logistic_regression	train
3	F1	0.950820	stacked_logistic_regression	train
4	AUROC	0.982353	stacked_logistic_regression	train

##################################
# Evaluating the stacked Logistic Regression model
# on the test set
##################################
stacked_logistic_regression_y_hat_test = stacked_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
stacked_logistic_regression_performance_test = model_performance_evaluation(y_test, stacked_logistic_regression_y_hat_test)
stacked_logistic_regression_performance_test['model'] = ['stacked_logistic_regression'] * 5
stacked_logistic_regression_performance_test['set'] = ['test'] * 5
print('Stacked Logistic Regression Model Performance on Test Data: ')
display(stacked_logistic_regression_performance_test)

Stacked Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.918367	stacked_logistic_regression	test
1	Precision	0.900000	stacked_logistic_regression	test
2	Recall	0.750000	stacked_logistic_regression	test
3	F1	0.818182	stacked_logistic_regression	test
4	AUROC	0.861486	stacked_logistic_regression	test

1.3.10 Model Selection

1. Among the formulated versions of the logistic regression model, the model which applied upsampling of the minority class using SMOTE was used as a base learner for the model stacking algorithm.
   • Accuracy = 0.9183
   • Precision = 0.9000
   • Recall = 0.7500
   • F1 Score = 0.8181
   • AUROC = 0.8614
2. Among the formulated versions of the decision tree model, the model which applied upsampling of the minority class using SMOTE was used as a base learner for the model stacking algorithm.
   • Accuracy = 0.8979
   • Precision = 0.7692
   • Recall = 0.8333
   • F1 Score = 0.8000
   • AUROC = 0.8761
3. Among the formulated versions of the random forest model, the model which applied upsampling of the minority class using SMOTE was used as a base learner for the model stacking algorithm.
   • Accuracy = 0.9387
   • Precision = 0.8461
   • Recall = 0.9167
   • F1 Score = 0.8800
   • AUROC = 0.9313
4. Among the formulated versions of the support vector machine model, the model which applied upsampling of the minority class using SMOTE was used as a base learner for the model stacking algorithm.
   • Accuracy = 0.8979
   • Precision = 0.8181
   • Recall = 0.7500
   • F1 Score = 0.7826
   • AUROC = 0.8479
5. The stacked logistic regression model comprised of the individual base learners demonstrated sufficient class discrmination:
   • Accuracy = 0.9183
   • Precision = 0.9000
   • Recall = 0.7500
   • F1 Score = 0.8181
   • AUROC = 0.8614
6. Comparing all results from the formulated base and stacked models formulated, the logistic regression model which applied class weights still demonstrated the best independent test model performance and was selected as the final model for classification.
   • Accuracy = 0.9387
   • Precision = 0.8461
   • Recall = 0.9167
   • F1 Score = 0.8800
   • AUROC = 0.9313

##################################
# Consolidating all the
# base and meta-learner
# model performance measures
##################################
base_meta_learner_performance_comparison = pd.concat([weighted_logistic_regression_performance_train, 
                                                      weighted_logistic_regression_performance_test,
                                                      upsampled_logistic_regression_performance_train, 
                                                      upsampled_logistic_regression_performance_test,
                                                      upsampled_decision_tree_performance_train, 
                                                      upsampled_decision_tree_performance_test,
                                                      upsampled_random_forest_performance_train, 
                                                      upsampled_random_forest_performance_test,
                                                      upsampled_support_vector_machine_performance_train, 
                                                      upsampled_support_vector_machine_performance_test,
                                                      stacked_logistic_regression_performance_train, 
                                                      stacked_logistic_regression_performance_test], 
                                                     ignore_index=True)
print('Consolidated Base and Meta Learner Model Performance on Train and Test Data: ')
display(base_meta_learner_performance_comparison)

Consolidated Base and Meta Learner Model Performance on Train and Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.894737	weighted_logistic_regression	train
1	Precision	0.707317	weighted_logistic_regression	train
2	Recall	1.000000	weighted_logistic_regression	train
3	F1	0.828571	weighted_logistic_regression	train
4	AUROC	0.929412	weighted_logistic_regression	train
5	Accuracy	0.938776	weighted_logistic_regression	test
6	Precision	0.846154	weighted_logistic_regression	test
7	Recall	0.916667	weighted_logistic_regression	test
8	F1	0.880000	weighted_logistic_regression	test
9	AUROC	0.931306	weighted_logistic_regression	test
10	Accuracy	0.964912	upsampled_logistic_regression	train
11	Precision	0.903226	upsampled_logistic_regression	train
12	Recall	0.965517	upsampled_logistic_regression	train
13	F1	0.933333	upsampled_logistic_regression	train
14	AUROC	0.965112	upsampled_logistic_regression	train
15	Accuracy	0.918367	upsampled_logistic_regression	test
16	Precision	0.900000	upsampled_logistic_regression	test
17	Recall	0.750000	upsampled_logistic_regression	test
18	F1	0.818182	upsampled_logistic_regression	test
19	AUROC	0.861486	upsampled_logistic_regression	test
20	Accuracy	0.921053	upsampled_decision_tree	train
21	Precision	0.763158	upsampled_decision_tree	train
22	Recall	1.000000	upsampled_decision_tree	train
23	F1	0.865672	upsampled_decision_tree	train
24	AUROC	0.947059	upsampled_decision_tree	train
25	Accuracy	0.897959	upsampled_decision_tree	test
26	Precision	0.769231	upsampled_decision_tree	test
27	Recall	0.833333	upsampled_decision_tree	test
28	F1	0.800000	upsampled_decision_tree	test
29	AUROC	0.876126	upsampled_decision_tree	test
30	Accuracy	0.991228	upsampled_random_forest	train
31	Precision	0.966667	upsampled_random_forest	train
32	Recall	1.000000	upsampled_random_forest	train
33	F1	0.983051	upsampled_random_forest	train
34	AUROC	0.994118	upsampled_random_forest	train
35	Accuracy	0.918367	upsampled_random_forest	test
36	Precision	0.900000	upsampled_random_forest	test
37	Recall	0.750000	upsampled_random_forest	test
38	F1	0.818182	upsampled_random_forest	test
39	AUROC	0.861486	upsampled_random_forest	test
40	Accuracy	0.973684	upsampled_support_vector_machine	train
41	Precision	0.906250	upsampled_support_vector_machine	train
42	Recall	1.000000	upsampled_support_vector_machine	train
43	F1	0.950820	upsampled_support_vector_machine	train
44	AUROC	0.982353	upsampled_support_vector_machine	train
45	Accuracy	0.897959	upsampled_support_vector_machine	test
46	Precision	0.818182	upsampled_support_vector_machine	test
47	Recall	0.750000	upsampled_support_vector_machine	test
48	F1	0.782609	upsampled_support_vector_machine	test
49	AUROC	0.847973	upsampled_support_vector_machine	test
50	Accuracy	0.973684	stacked_logistic_regression	train
51	Precision	0.906250	stacked_logistic_regression	train
52	Recall	1.000000	stacked_logistic_regression	train
53	F1	0.950820	stacked_logistic_regression	train
54	AUROC	0.982353	stacked_logistic_regression	train
55	Accuracy	0.918367	stacked_logistic_regression	test
56	Precision	0.900000	stacked_logistic_regression	test
57	Recall	0.750000	stacked_logistic_regression	test
58	F1	0.818182	stacked_logistic_regression	test
59	AUROC	0.861486	stacked_logistic_regression	test

##################################
# Consolidating all the F1 score
# model performance measures
##################################
base_meta_learner_performance_comparison_F1 = base_meta_learner_performance_comparison[base_meta_learner_performance_comparison['metric_name']=='F1']
base_meta_learner_performance_comparison_F1_train = base_meta_learner_performance_comparison_F1[base_meta_learner_performance_comparison_F1['set']=='train'].loc[:,"metric_value"]
base_meta_learner_performance_comparison_F1_test = base_meta_learner_performance_comparison_F1[base_meta_learner_performance_comparison_F1['set']=='test'].loc[:,"metric_value"]

##################################
# Combining all the F1 score
# model performance measures
# between train and test sets
##################################
base_meta_learner_performance_comparison_F1_plot = pd.DataFrame({'train': base_meta_learner_performance_comparison_F1_train.values,
                                                                 'test': base_meta_learner_performance_comparison_F1_test.values},
                                                                index=base_meta_learner_performance_comparison_F1['model'].unique())
base_meta_learner_performance_comparison_F1_plot

	train	test
weighted_logistic_regression	0.828571	0.880000
upsampled_logistic_regression	0.933333	0.818182
upsampled_decision_tree	0.865672	0.800000
upsampled_random_forest	0.983051	0.818182
upsampled_support_vector_machine	0.950820	0.782609
stacked_logistic_regression	0.950820	0.818182

##################################
# Plotting all the F1 score
# model performance measures
# between train and test sets
##################################
base_meta_learner_performance_comparison_F1_plot = base_meta_learner_performance_comparison_F1_plot.plot.barh(figsize=(10, 6))
base_meta_learner_performance_comparison_F1_plot.set_xlim(0.00,1.00)
base_meta_learner_performance_comparison_F1_plot.set_title("Model Comparison by F1 Score Performance on Test Data")
base_meta_learner_performance_comparison_F1_plot.set_xlabel("F1 Score Performance")
base_meta_learner_performance_comparison_F1_plot.set_ylabel("Base and Meta Learner Model")
base_meta_learner_performance_comparison_F1_plot.grid(False)
base_meta_learner_performance_comparison_F1_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in base_meta_learner_performance_comparison_F1_plot.containers:
    base_meta_learner_performance_comparison_F1_plot.bar_label(container, fmt='%.5f', padding=-50)

##################################
# Consolidating all score
# model performance measures
##################################
base_meta_learner_performance_comparison_Accuracy_test = base_meta_learner_performance_comparison[(base_meta_learner_performance_comparison['set']=='test') & (base_meta_learner_performance_comparison['metric_name']=='Accuracy')].loc[:,"metric_value"]
base_meta_learner_performance_comparison_Precision_test = base_meta_learner_performance_comparison[(base_meta_learner_performance_comparison['set']=='test') & (base_meta_learner_performance_comparison['metric_name']=='Precision')].loc[:,"metric_value"]
base_meta_learner_performance_comparison_Recall_test = base_meta_learner_performance_comparison[(base_meta_learner_performance_comparison['set']=='test') & (base_meta_learner_performance_comparison['metric_name']=='Recall')].loc[:,"metric_value"]
base_meta_learner_performance_comparison_F1_test = base_meta_learner_performance_comparison[(base_meta_learner_performance_comparison['set']=='test') & (base_meta_learner_performance_comparison['metric_name']=='F1')].loc[:,"metric_value"]
base_meta_learner_performance_comparison_AUROC_test = base_meta_learner_performance_comparison[(base_meta_learner_performance_comparison['set']=='test') & (base_meta_learner_performance_comparison['metric_name']=='AUROC')].loc[:,"metric_value"]

##################################
# Combining all the score
# model performance measures
# between train and test sets
##################################
base_meta_learner_performance_comparison_all_plot = pd.DataFrame({'accuracy': base_meta_learner_performance_comparison_Accuracy_test.values,
                                                                  'precision': base_meta_learner_performance_comparison_Precision_test.values,
                                                                  'recall': base_meta_learner_performance_comparison_Recall_test.values,
                                                                  'f1': base_meta_learner_performance_comparison_F1_test.values,
                                                                  'auroc': base_meta_learner_performance_comparison_AUROC_test.values},
                                                                index=base_meta_learner_performance_comparison['model'].unique())
base_meta_learner_performance_comparison_all_plot

	accuracy	precision	recall	f1	auroc
weighted_logistic_regression	0.938776	0.846154	0.916667	0.880000	0.931306
upsampled_logistic_regression	0.918367	0.900000	0.750000	0.818182	0.861486
upsampled_decision_tree	0.897959	0.769231	0.833333	0.800000	0.876126
upsampled_random_forest	0.918367	0.900000	0.750000	0.818182	0.861486
upsampled_support_vector_machine	0.897959	0.818182	0.750000	0.782609	0.847973
stacked_logistic_regression	0.918367	0.900000	0.750000	0.818182	0.861486

##################################
# Plotting all the score
# model performance measures
# between train and test sets
##################################
base_meta_learner_performance_comparison_all_plot = base_meta_learner_performance_comparison_all_plot.plot.barh(figsize=(10, 9),width=0.90)
base_meta_learner_performance_comparison_all_plot.set_xlim(0.00,1.00)
base_meta_learner_performance_comparison_all_plot.set_title("Model Comparison by Score Performance on Test Data")
base_meta_learner_performance_comparison_all_plot.set_xlabel("Score Performance")
base_meta_learner_performance_comparison_all_plot.set_ylabel("Base and Meta Learner Model")
base_meta_learner_performance_comparison_all_plot.grid(False)
base_meta_learner_performance_comparison_all_plot.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
for container in base_meta_learner_performance_comparison_all_plot.containers:
    base_meta_learner_performance_comparison_all_plot.bar_label(container, fmt='%.5f', padding=-50)

##################################
# Plotting the confusion matrices
# for all the Support Vector Machine models
##################################
classifiers = {"upsampled_logistic_regression": upsampled_logistic_regression,
               "upsampled_decision_tree": upsampled_decision_tree,
               "upsampled_random_forest": upsampled_random_forest,
               "upsampled_support_vector_machine": upsampled_support_vector_machine,
               "stacked_logistic_regression": stacked_logistic_regression,
               "weighted_logistic_regression": weighted_logistic_regression,}

fig, axes = plt.subplots(2, 3, figsize=(20, 10))
axes = axes.ravel()
for i, (key, classifier) in enumerate(classifiers.items()):
    y_pred = classifier.predict(X_test)
    cf_matrix = confusion_matrix(y_test, y_pred)
    disp = ConfusionMatrixDisplay(cf_matrix)
    disp.plot(ax=axes[i], xticks_rotation=0)
    disp.ax_.grid(False)
    disp.ax_.set_title(key)
    disp.im_.colorbar.remove()

fig.colorbar(disp.im_, ax=axes)
plt.show()

1.3.11 Model Presentation

1.3.11.1 Odds Ratios

1. The most important predictors in the model ranked by their Absolute Coefficient Value and Estimated Odds Ratio for a CANRAT=HIGH Prediction were listed as follows.
   • EPISCO: Model.Coefficient=+1.136, Odds.Ratio=3.114
   • GDPCAP: Model.Coefficient=+0.596, Odds.Ratio=1.815
   • DTHCMD: Model.Coefficient=-0.534, Odds.Ratio=0.586
   • LIFEXP: Model.Coefficient=+0.473, Odds.Ratio=1.604
   • TUBINC: Model.Coefficient=-0.412, Odds.Ratio=0.662
   • HDICAT_VH: Model.Coefficient=+0.268, Odds.Ratio=1.308
   • CO2EMI: Model.Coefficient=-0.151, Odds.Ratio=0.860
   • URBPOP: Model.Coefficient=+0.094, Odds.Ratio=1.098

##################################
# Reformulating the weighted Logistic Regression model
# as the final classification model
# with the optimal hyperparameters
##################################
final_model = LogisticRegression(C=1.0,
                                 class_weight={0: 0.25, 1: 0.75},
                                 solver='liblinear',
                                 penalty= 'l2',
                                 max_iter=500,
                                 random_state=88888888)
final_model.fit(X_train, y_train)

LogisticRegression(class_weight={0: 0.25, 1: 0.75}, max_iter=500,
                   random_state=88888888, solver='liblinear')

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

##################################
# Gathering the model coefficients
# and the estimated log-odds 
# of the weighted Logistic Regression model
##################################
final_model_coefficient = pd.DataFrame(zip(X_train.columns, 
                                           final_model.coef_[0].tolist(),
                                           np.exp(final_model.coef_)[0].tolist()),
                                       columns=['model_predictor','model_coefficient','odds_ratio'])
display(final_model_coefficient)

	model_predictor	model_coefficient	odds_ratio
0	URBPOP	0.093942	1.098496
1	LIFEXP	0.472572	1.604115
2	TUBINC	-0.412017	0.662313
3	DTHCMD	-0.534044	0.586229
4	CO2EMI	-0.150646	0.860152
5	GDPCAP	0.596013	1.814868
6	EPISCO	1.135875	3.113897
7	HDICAT_VH	0.268438	1.307920

1.3.11.2 Shapley Additive Explanations

1. The most important predictors in the model ranked by their Mean Shap Value and Feature Impact to CANRAT=HIGH Prediction were listed as follows.
   • EPISCO: Mean.Shap.Value=1.00, Feature.Impact=Positive
   • GDPCAP: Model.Coefficient=0.48, Feature.Impact=Positive
   • DTHCMD: Model.Coefficient=0.46, Feature.Impact=Negative
   • LIFEXP: Model.Coefficient=0.38, Feature.Impact=Positive
   • TUBINC: Model.Coefficient=0.36, Feature.Impact=Negative
   • HDICAT_VH: Model.Coefficient=0.13, Feature.Impact=Positive
   • CO2EMI: Model.Coefficient=0.13, Feature.Impact=Negative
   • URBPOP: Model.Coefficient=0.08, Feature.Impact=Positive

##################################
# Setting up the primary explainer interface
# for the SHAP library using the 
# weighted Logistic Regression model
##################################
final_model_explainer = shap.Explainer(final_model, X_train)

##################################
# Gathering up the SHAP values
# for the train set
##################################
final_model_train_shap_values = final_model_explainer(X_train)

##################################
# Gathering up the SHAP values
# for the test set
##################################
final_model_test_shap_values = final_model_explainer(X_test)

##################################
# Formulating the bar plot
# of the SHAP values using the train set
# to estimate global feature importance
##################################
shap.plots.bar(final_model_train_shap_values, show=False)
plt.xlim([0, 1])
plt.show()

##################################
# Formulating the bar plot
# of the SHAP values using the test set
# to estimate global feature importance
##################################
shap.plots.bar(final_model_test_shap_values, show=False)
plt.xlim([0, 1])
plt.show()

##################################
# Formulating the beeswarm plot
# of the SHAP values using the train set
# to estimate the feature impact
# on model predictions
##################################
shap.plots.beeswarm(final_model_train_shap_values, show=False, plot_size=(10, 6))
plt.xlim([-3, 3])
plt.show()

##################################
# Formulating the beeswarm plot
# of the SHAP values using the test set
# to estimate the feature impact
# on model predictions
##################################
shap.plots.beeswarm(final_model_test_shap_values, show=False, plot_size=(10, 6))
plt.xlim([-3, 3])
plt.show()

##################################
# Formulating the heatmap plot
# of the SHAP values using the train set
# to estimate the observation impact
# on model predictions
##################################
shap.plots.heatmap(final_model_train_shap_values)

##################################
# Formulating the heatmap plot
# of the SHAP values using the train set
# to estimate the observation impact
# on model predictions
##################################
shap.plots.heatmap(final_model_test_shap_values)

##################################
# Formulating the dependence plot
# of the SHAP values using the train set
# for the most important feature
# as evaluated to the rest of the features
##################################
shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="GDPCAP")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="DTHCMD")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="LIFEXP")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="TUBINC")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="CO2EMI")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="HDICAT_VH")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_train), X_train, interaction_index="URBPOP")

##################################
# Formulating the dependence plot
# of the SHAP values using the test set
# for the most important feature
# as evaluated to the rest of the features
##################################
shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="GDPCAP")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="DTHCMD")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="LIFEXP")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="TUBINC")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="CO2EMI")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="HDICAT_VH")

shap.dependence_plot('EPISCO', final_model_explainer.shap_values(X_test), X_test, interaction_index="URBPOP")

2. Summary

A logistic regression model applied with L2 regularization and class weights provided a set of robust and reliable predictions for countries belonging to the high cancer rate group – predominantly characterized by factors related to social development (life expectancy, human development index), economic (GDP per capita, urban population), healthcare delivery (death by communicable disease, tuberculosis incidence) and environmental (environmental protection index, CO2 emission) factors. The key drivers identified for high cancer rate levels ranked by feature importance with their conditioned effects indicated were given as follows:

• Environmental protection index (+)
• GDP per capita (+)
• Death by communicable disease (-)
• Life expectancy (+)
• Tuberculosis incidence (-)
• Human development index (+)
• CO2 emission (-)
• Urban population (+)

Overall, industrialized economies tend to belong to the cluster of countries with higher cancer rates. While progressiveness may not inherently imply more cancer prevalence, these countries potentially have advanced healthcare systems with robust screening and diagnostic capabilities. This can result in more thorough and accurate detection of cancer cases, leading to higher reported incidence rates. Improved reporting mechanisms contribute to a better understanding of the true burden of cancer. Progressive countries also often have higher life expectancies, resulting in older populations. Cancer incidence tends to increase with age, so countries with aging populations may experience higher overall cancer rates. Additionally, industrialization and urbanization, often associated with progressiveness, may lead to increased exposure to environmental pollutants and carcinogens. Certain industrial activities, pollution levels, and occupational exposures can contribute to higher cancer rates. Given these observations, the relationship between progressiveness and cancer rates is complex, and multiple factors contribute to observed patterns. It is essential to consider the specific context of each country and conduct detailed analyses to understand the underlying reasons for variations in cancer incidence.

• From an initial dataset comprised of 177 observations and 21 predictors, an optimal subset of 163 observations and 8 predictors representing social development, economic, healthcare delivery and environmental factors were determined after conducting data quality assessment and feature selection, excluding cases or variables noted with irregularities and applying preprocessing operations most suitable for the downstream analysis.

• Multiple classification modelling algorithms with various hyperparameter combinations were formulated using Logistic Regression, Decision Tree, Random Forest and Support Vector Machine. Class imbalance treatment including Class Weights, Upsampling with Synthetic Minority Oversampling Technique (SMOTE) and Downsampling with Condensed Nearest Neighbors (CNN) were implemented. Ensemble Learning Using Model Stacking was additionally explored. The best model with optimized hyperparameters from each algorithm were determined through internal resampling validation using 5-Fold Cross Validation with F1 Score used as the primary performance metric among Accuracy, Precision, Recall and Area Under the Receiver Operating Characterisng Curve (AUROC). All candidate models were compared based on internal and external validation performance.

• The final model selected among candidates used Logistic Regression Model defined by an L2 Regularization and Class Weights with optimal hyperparameters: weights associated with classes (class_weight={0;LOW: 0.25, 1;HIGH: 0.75}), inverse of regularization strength (C=1), regularization (penalty=L2), algorithm used in the optimization problem (solver=liblinear) and maximum number of iterations taken for the solvers to converge (max_iter=500). This model demonstrated the best externally validated F1 Score, AUROC, Precision, Recall and Accuracy (F1 Score=0.88, AUROC=0.93, Precision=0.85, Recall=0.92, Accuracy=0.94) with no excessive overfitting comparing the external and apparent validation metrics .

• Post-hoc exploration of the model results involved model-specific (Odds Ratios) and model-agnostic (Shapley Additive Explanations) methods. Both methods were consistent in ranking Environmental protection index, GDP per capita, Death by communicable disease, Life expectancy, Tuberculosis incidence, Human development index, CO2 emission and Urban population as the most important features by importance. These results helped provide insights on the significance, contribution and effect of the various predictors to model prediction.

3. References

• [Book] Data Preparation for Machine Learning: Data Cleaning, Feature Selection, and Data Transforms in Python by Jason Brownlee
• [Book] Feature Engineering and Selection: A Practical Approach for Predictive Models by Max Kuhn and Kjell Johnson
• [Book] Feature Engineering for Machine Learning by Alice Zheng and Amanda Casari
• [Book] Applied Predictive Modeling by Max Kuhn and Kjell Johnson
• [Book] Data Mining: Practical Machine Learning Tools and Techniques by Ian Witten, Eibe Frank, Mark Hall and Christopher Pal
• [Book] Data Cleaning by Ihab Ilyas and Xu Chu
• [Book] Data Wrangling with Python by Jacqueline Kazil and Katharine Jarmul
• [Book] Regression Modeling Strategies by Frank Harrell
• [Book] Ensemble Methods for Machine Learning by Gautam Kunapuli
• [Book] Imbalanced Classification with Python: Better Metrics, Balance Skewed Classes, Cost-Sensitive Learning by Jason Brownlee
• [Python Library API] NumPy by NumPy Team
• [Python Library API] pandas by Pandas Team
• [Python Library API] seaborn by Seaborn Team
• [Python Library API] matplotlib.pyplot by MatPlotLib Team
• [Python Library API] itertools by Python Team
• [Python Library API] operator by Python Team
• [Python Library API] sklearn.experimental by Scikit-Learn Team
• [Python Library API] sklearn.impute by Scikit-Learn Team
• [Python Library API] sklearn.linear_model by Scikit-Learn Team
• [Python Library API] sklearn.preprocessing by Scikit-Learn Team
• [Python Library API] scipy by SciPy Team
• [Python Library API] sklearn.tree by Scikit-Learn Team
• [Python Library API] sklearn.ensemble by Scikit-Learn Team
• [Python Library API] sklearn.svm by Scikit-Learn Team
• [Python Library API] sklearn.metrics by Scikit-Learn Team
• [Python Library API] sklearn.model_selection by Scikit-Learn Team
• [Python Library API] imblearn.over_sampling by Imbalanced-Learn Team
• [Python Library API] imblearn.under_sampling by Imbalanced-Learn Team
• [Article] Step-by-Step Exploratory Data Analysis (EDA) using Python%20with,distributions%20using%20Python%20programming%20language.) by Malamahadevan Mahadevan (Analytics Vidhya)
• [Article] Exploratory Data Analysis in Python — A Step-by-Step Process by Andrea D'Agostino (Towards Data Science)
• [Article] Exploratory Data Analysis with Python by Douglas Rocha (Medium)
• [Article] 4 Ways to Automate Exploratory Data Analysis (EDA) in Python by Abdishakur Hassan (BuiltIn)
• [Article] 10 Things To Do When Conducting Your Exploratory Data Analysis (EDA) by Alifia Harmadi (Medium)
• [Article] How to Handle Missing Data with Python by Jason Brownlee (Machine Learning Mastery)
• [Article] Statistical Imputation for Missing Values in Machine Learning by Jason Brownlee (Machine Learning Mastery)
• [Article] Imputing Missing Data with Simple and Advanced Techniques by Idil Ismiguzel (Towards Data Science)
• [Article] Missing Data Imputation Approaches | How to handle missing values in Python by Selva Prabhakaran (Machine Learning +)
• [Article] Master The Skills Of Missing Data Imputation Techniques In Python(2022) And Be Successful by Mrinal Walia (Analytics Vidhya)
• [Article] How to Preprocess Data in Python by Afroz Chakure (BuiltIn)
• [Article] Easy Guide To Data Preprocessing In Python by Ahmad Anis (KDNuggets)
• [Article] Data Preprocessing in Python by Tarun Gupta (Towards Data Science)
• [Article] Data Preprocessing using Python by Suneet Jain (Medium)
• [Article] Data Preprocessing in Python by Abonia Sojasingarayar (Medium)
• [Article] Data Preprocessing in Python by Afroz Chakure (Medium)
• [Article] Detecting and Treating Outliers | Treating the Odd One Out! by Harika Bonthu (Analytics Vidhya)
• [Article] Outlier Treatment with Python by Sangita Yemulwar (Analytics Vidhya)
• [Article] A Guide to Outlier Detection in Python by Sadrach Pierre (BuiltIn)
• [Article] How To Find Outliers in Data Using Python (and How To Handle Them) by Eric Kleppen (Career Foundry)
• [Article] Statistics in Python — Collinearity and Multicollinearity by Wei-Meng Lee (Towards Data Science)
• [Article] Understanding Multicollinearity and How to Detect it in Python by Terence Shin (Towards Data Science)
• [Article] A Python Library to Remove Collinearity by Gianluca Malato (Your Data Teacher)
• [Article] 8 Best Data Transformation in Pandas by Tirendaz AI (Medium)
• [Article] Data Transformation Techniques with Python: Elevate Your Data Game! by Siddharth Verma (Medium)
• [Article] Data Scaling with Python by Benjamin Obi Tayo (KDNuggets)
• [Article] How to Use StandardScaler and MinMaxScaler Transforms in Python by Jason Brownlee (Machine Learning Mastery)
• [Article] Feature Engineering: Scaling, Normalization, and Standardization by Aniruddha Bhandari (Analytics Vidhya)
• [Article] How to Normalize Data Using scikit-learn in Python by Jayant Verma (Digital Ocean)
• [Article] What are Categorical Data Encoding Methods | Binary Encoding by Shipra Saxena (Analytics Vidhya)
• [Article] Guide to Encoding Categorical Values in Python by Chris Moffitt (Practical Business Python)
• [Article] Categorical Data Encoding Techniques in Python: A Complete Guide by Soumen Atta (Medium)
• [Article] Categorical Feature Encoding Techniques by Tara Boyle (Medium)
• [Article] Ordinal and One-Hot Encodings for Categorical Data by Jason Brownlee (Machine Learning Mastery)
• [Article] Hypothesis Testing with Python: Step by Step Hands-On Tutorial with Practical Examples by Ece Işık Polat (Towards Data Science)
• [Article] 17 Statistical Hypothesis Tests in Python (Cheat Sheet) by Jason Brownlee (Machine Learning Mastery)
• [Article] A Step-by-Step Guide to Hypothesis Testing in Python using Scipy by Gabriel Rennó (Medium)
• [Article] How to Evaluate Classification Models in Python: A Beginner's Guide by Sadrach Pierre (BuiltIn)
• [Article] Machine Learning Classifiers Comparison with Python by Roberto Salazar (Towards Data Science)
• [Article] Top 6 Machine Learning Algorithms for Classification by Destin Gong (Towards Data Science)
• [Article] Metrics For Evaluating Machine Learning Classification Models by Cory Maklin (Towards Data Science)
• [Article] Evaluation Metrics for Classification Problems with Implementation in Python by Venu Gopal Kadamba (Medium)
• [Article] Tour of Evaluation Metrics for Imbalanced Classification by Jason Brownlee (Machine Learning Mastery)
• [Article] Metrics To Evaluate Machine Learning Algorithms in Python by Jason Brownlee (Machine Learning Mastery)
• [Article] How To Compare Machine Learning Algorithms in Python with scikit-learn by Jason Brownlee (Machine Learning Mastery)
• [Article] How to Deal With Imbalanced Classification and Regression Data by Prince Canuma (Neptune.AI)
• [Article] Random Oversampling and Undersampling for Imbalanced Classification by Jason Brownlee (Machine Learning Mastery)
• [Article] How to Handle Imbalance Data and Small Training Sets in ML by Ege Hosgungor (Towards Data Science)
• [Article] Class Imbalance Strategies — A Visual Guide with Code by Travis Tang (Towards Data Science)
• [Article] Machine Learning: How to Handle Class Imbalance by Ken Hoffman (Medium)
• [Article] Handling Class Imbalance in Machine Learning by Okan Yenigün (Medium)
• [Article] Undersampling Algorithms for Imbalanced Classification by Jason Brownlee (Machine Learning Mastery)
• [Article] Condensed Nearest Neighbor Rule Undersampling (CNN) & TomekLinks by Rupak Roy (Medium)
• [Article] CNN (Condensed Nearest Neighbors) by Abhishek (Medium)
• [Article] Synthetic Minority Over-sampling TEchnique (SMOTE) by Cory Maklin (Medium)
• [Article] SMOTE for Imbalanced Classification with Python by Swastik Satpathy (Analytics Vidhya)
• [Article] An Introduction to SMOTE%20is%20an%20oversampling%20method%20of,a%20point%20along%20that%20line.) by Abid Ali Awan (KD Nuggets)
• [Article] A Comprehensive Guide to Ensemble Learning (with Python codes) by Aishwarya Singh (Analytics Vidhya)
• [Article] Stacked Ensembles — Improving Model Performance on a Higher Level by Yenwee Lim (Towards Data Science)
• [Article] Stacking to Improve Model Performance: A Comprehensive Guide on Ensemble Learning in Python by Brijesh Soni (Medium)
• [Article] Stacking Ensemble Machine Learning With Python by Jason Brownlee (Machine Learning Mastery)
• [Publication] Data Quality for Machine Learning Tasks by Nitin Gupta, Shashank Mujumdar, Hima Patel, Satoshi Masuda, Naveen Panwar, Sambaran Bandyopadhyay, Sameep Mehta, Shanmukha Guttula, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’21: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining)
• [Publication] Overview and Importance of Data Quality for Machine Learning Tasks by Abhinav Jain, Hima Patel, Lokesh Nagalapatti, Nitin Gupta, Sameep Mehta, Shanmukha Guttula, Shashank Mujumdar, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)
• [Publication] Multiple Imputation of Discrete and Continuous Data by Fully Conditional Specification by Stef van Buuren (Statistical Methods in Medical Research)
• [Publication] Mathematical Contributions to the Theory of Evolution: Regression, Heredity and Panmixia by Karl Pearson (Royal Society)
• [Publication] A New Family of Power Transformations to Improve Normality or Symmetry by In-Kwon Yeo and Richard Johnson (Biometrika)
• [Publication] The Probable Error of the Mean by Student (Biometrika)
• [Publication] On the Criterion That a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such That It can Be Reasonably Supposed to Have Arisen From Random Sampling by Karl Pearson (Philosophical Magazine)
• [Publication] The Origins of Logistic Regression by JS Cramer (Econometrics eJournal)
• [Publication] Classification and Regression Trees by Leo Breiman, Jerome Friedman, Richard Olshen and Charles Stone (Computer Science)
• [Publication] Random Forest by Leo Breiman (Machine Learning)
• [Publication] A Training Algorithm for Optimal Margin Classifiers by Bernhard Boser, Isabelle Guyon and Vladimir Vapnik (Proceedings of the Fifth Annual Workshop on Computational Learning Theory)
• [Publication] SMOTE: Synthetic Minority Over-Sampling Technique by Nitesh Chawla, Kevin Bowyer, Lawrence Hall and Philip Kegelmeyer (Journal of Artificial Intelligence Research)
• [Publication] The Condensed Nearest Neighbor Rule by Peter Hart (IEEE Transactions on Information Theory)

---

from IPython.display import display, HTML
display(HTML("<style>.rendered_html { font-size: 15px; font-family: 'Trebuchet MS'; }</style>"))